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CARNEGIE INSTITUTE 

OF TECHNOLOGY 

LIBRARY 




PRESENTED BY 

Dr*Lloyd L. Dines 



A COURSE IN MATHEMATICAL ANALYSIS 

FUNCTIONS OF 
A COMPLEX VARIABLE 

BEING PART I OF VOLUME II 

BY 

EDOUARD GOURSAT 

PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF PARIS 

TRANSLATED BY 

EARLE RAYMOND HEDRICK 

PROFESSOR OF MATHETtfATICS, THE UNIVERSITY OF MISSOURI 

AND 

OTTO DUNKEL 

ASSOCIATE PROFESSOR OF MATHEMATICS, WASHINGTON UNIVERSITY 



GINN AND COMPANY 

BOSTON NEW YORK - CHICAGO LONDON 
ATLANTA DALLAS COLUMBUS SAN FRANCISCO 



COPYRIGHT, 1916, BY 
EARLE RAYMOND HEDRICK AND OTTO DUNKEL 



ALL BIGHTS RESERVED 

PRINTED IN THE UNITED STATES OF AMERICA 
226,11 



fltfttnaum 



GINN AND COMPANY . PRO 
PRIETORS BOSTON USA 



AUTHOR'S PREFACE SECOND FRENCH EDITION 

The first part of this volume has undeigone only slight changes, 
while the lather important modifications that have been made 
appear only m the last chapters 

In the first edition I was able to devote but a few pages to par- 
tial differential equations of the second ordei and to the calculus 
of variations In ordei to present in a less summary manner such 
broad subjects, I have concluded to defer them to a third volume, 
which will contain also a sketch of the recent theory of integral 
equations The suppression of the last chapter has enabled me to 
make some additions, of which the most important relate to linear 
differential equations and to partial differential equations of the 

first Oldei E GOTJESAT 



TRANSLATORS' PREFACE 

As the title indicates, the present volume is a translation of the 
first half of the second volume of Goursat's "Cours d' Analyse " The 
decision to publish the translation in two parts is due to the evi- 
dent adaptation of these two portions to the introductory courses in 
American colleges and universities in the theory of functions and 
in differential equations, respectively. 

After the cordial reception given to the translation of Goursat's 
first volume, the continuation was assured. That it has been 
delayed so long was due, in the first instance., to our desire to await 
the appearance of the second edition of the second volume in 
French The advantage in doing so will be obvious to those who 
have observed the radical changes made in the second (French) 
edition of the second volume Volume I was not altered so radi- 
cally, so that the present English translation of that volume may be 
used conveniently as a companion to this ; but references are given 
here to both editions of the first volume, to avoid any possible 
difficulty in this connection. 

Our thanks are due to Professor Goursat, who has kindly given 
us his permission to make this translation, and has approved of the 
plan of publication in two parts He has also seen all proofs in 
English and has approved a few minor alterations made in transla- 
tion as well as the translators' notes. The responsibility for the 

latter rests, however, with the translators. 

E. R. HEDRICK 

OTTO DTJNKEL 



CONTENTS 



PAGE 
CHAPTER I ELEMENTS OF THE THEORY . 3 

I GENERAL PRINCIPLES ANALYTIC FUNCTIONS 3 

1 Definitions . . 3 

2 Continuous functions of a complex variable 6 

3 Analytic functions . . 7 

4 Functions analytic throughout a region . 11 

5 Rational functions . .12 

6 Ceitam irrational functions . 13 

7 Single-valued and multiple-valued functions 17 

II POWER SERIES WITH COMPLEX TERMS ELEMENTARY 

TRANSCENDENTAL FUNCTIONS 18 

8 Circle of conveigence , . 18 

9 Double seiies . . 21 

10 Development of an infinite pioduct in power series 22 

11 The exponential function . 23 
12. Trigonometric functions . . 26 

13 Logarithms . . 28 

14 Inverse functions arc sin z, arc tan s 30 

15 Application to the integral calculus . 33 

16 Decomposition of a rational function of sin z and cos z into 

simple elements . . 35 

17. Expansion of Log (1 + z) . .38 

18 Extension of the binomial formula . .40 

III CONFORMAL REPRESENTATION . . 42 

19 Geometric interpretation of the derivative . 42 

20. Conformal transformations in general .... 45 

21. Conformal representation of one plane on another plane . . 48 

22 Riemann's theorem ... 50 

23 Geographic maps , . .52 

24 Isothermal curves .... . . 54 

EXERCISES 56 



Viii CONTENTS 

PAGE 

CHAPTER II THE GENERAL THEORY OF ANALYTIC FUNC- 
TIONS ACCORDING TO CAUCHY ... 60 

I. DEFINITE INTEGRALS TAKEN BETWEEN IMAGINARY LIMITS 60 

25 Definitions and general principles . 60 

26 Change of variables 62 

27 The formulae of Weierstrass and Daiboux 64 

28 Integrals taken along a closed cuive . 66 

31 Generalization of the formulae of the integral calculus , 72 

32 Another proof of the preceding results . . 74 

II CAUCHY'S INTEGRAL TAYLOR'S AND LAURENT'S SERIES 

SINGULAR POINTS EESIDUES . 75 

33 The fundamental formula 75 

34 Morera's theorem . 78 

35 Taylor's series 78 

36 Liouville's theorem 81 

37 Laurent's series . 81 

38 Other series . .84 

39 Series of analytic functions 86 

40 Poles ... . ,88 

41 Functions analytic except for poles . 90 

42 Essentially singular points . 01 

43 Residues . 04 

III. APPLICATIONS OF THE GENERAL THEOREMS , 05 

44 Introductory remarks .... . 05 

45 Evaluation of elementary definite integrals . 06 

46 Various definite integrals ... .07 

47 Evaluation of T( T(l-^) . . 100 

48 Application to functions analytic except for poles 101 

49 Application to the theory of equations ... . 103 

50 Jensen's formula . .... 104 
51. Lagrange's formula . ... 106 

52 Study of functions for infinite values of the variable 109 

IV. PERIODS OF DEFINITE INTEGRALS . . ... 112 

53 Polar periods . . . . Hg 

54. A study of the integral J^dz/^/l z* 114 

55. Periods of hyperelliptic integrals . .... 116 
56 Periods of elliptic integrals of the first kind 120 

EXERCISES 122 



CONTENTS ix 

PAGE 

CHAPTER III SINGLE-VALUED ANALYTIC FUNCTIONS . 127 

L WEIERSTRASS'S PRIMARY FUNCTIONS. MITTAG-LEFFLER'S 

THEOREM . 127 

57 Expression of an integial function as a product of primary 

functions 127 

58 The class of an integial function . 132 

59 Single-valued analytic functions with a finite number of 

singular points . . . 132 

60. Single-valued analytic functions with an infinite number of 

singular points . . .... 134 

61. Mittag-Leffler's theorem . . . 134 

62. Certain special cases . . 137 
63 Cauchy's method . . 139 
64. Expansion of ctn x and of sin x . 142 

II. DOUBLY PERIODIC FUNCTIONS ELLIPTIC FUNCTIONS 145 

65 Periodic functions Expansion in series 145 

66 Impossibility of a single-valued analytic function with 

three periods . . . 147 

67 Doubly periodic functions . . 149 

68 Elliptic functions General properties . . 150 

69 The fimctionp(w) . . . . 154 

70 The algebraic relation between p(u) and p'(u) . 158 

71 The function (u) . . 159 

72 The f unction <r(w) . . .162 

73 General expressions for elliptic functions . 163 

74 Addition formulae . 166 

75. Integration of elliptic functions . . 168 

76 The function . 170 

III. INVERSE FUNCTIONS. CURVES OF .DEFICIENCY ONE 172 

77. Relations between the penods and the invariants . . 172 

78 The inverse function to the elliptic integral of the first kind 174 

79. A new definition of p (u) by means of the invariants . . . 182 

80 Application to cubics in a plane . . . . 184 

81. General formulae for parameter representation . 187 

82 Curves of deficiency one .... . . 191 

EXERCISES . . . .... . .193 

CHAPTER IV. ANALYTIC EXTENSION 196 

L DEFINITION OF AN ANALYTIC FUNCTION BY MEANS OF 

ONE OF ITS ELEMENTS 196 

83 Introduction to analytic extension 196 

84. New definition of analytic functions 199 



x CONTENTS 

PAGE 

85 Singular points . . . 204 

86 Geneial pioblem . . 206 

II NATURAL BOUNDARIES CUTS 208 

87 Smgulai lines Natuial boundanes 208 

88 Examples 211 

89 Singularities of analytical expiessions 213 

90 Heinute's foimula . 215 

EXERCISES . . 217 
CHAPTER V ANALYTIC FUNCTIONS OF SEVERAL VARIABLES 219 

I GENERAL PROPERTIES 219 

91 Definitions . 219 

92 Associated cucles of convergence 220 

93 Double integrals 222 
94. Extension ot Cauchy's theoiems 225 

95 Functions repiesentod by definite integrals . 227 

96 Application to the F function . 229 

97 Analytic extension of a function of two variables 231 

II. IMPLICIT FUNCTIONS A.LGEBRAIC FUNCTIONS 232 

98 Weiersti ass's tlieoiem 232 

99 Critical points . . .236 

100 Algebraic functions . , 240 

101 Abehan integrals . . 243 

102 Abel's theoiem . 244 

103 Application to hyper elliptic integrals . 247 

104 Extension of Lag range's formula . 250 

EXERCISES , 2 r >2 

INDEX . 253 



A COURSE IN 
MATHEMATICAL ANALYSIS 

VOLUME II. PART I 



THEOEY OF FUNCTIONS OF A 
COMPLEX VARIABLE 

CHAPTER I 

ELEMENTS OF THE THEORY 
I GENERAL PRINCIPLES ANALYTIC FUNCTIONS 

1. Definitions. An ^ma,g^nary quantity, or complex quantity ', is any 
expression of the foim a + bi where a and b are any two real num- 
bers whatever and i is a special symbol which has been introduced 
in order to generalize algebra. Essentially a complex quantity is 
nothing but a system of two real numbers arranged in a certain 
order Although such expressions as a -h bi have in themselves no 
concrete meaning whatever, we agree to apply to them the ordinary 
rules of algebra, with the additional convention that i* shall be 
replaced throughout by 1 

Two complex quantities a -f- bi and a' + b'i are said to be equal if 
a = a' and b = b' The sum of two complex quantities a + bi and 
c + di is a symbol of the same form a 4- o + (b + d)t, the differ- 
ence a + bi (0-f di) is equal to a c + (b d)L To find the 
product of a + bi and c + di we carry out the multiplication accord- 
ing to the usual rules for algebraic multiplication, replacing i z by 
1, obtaining thus 

(a 4. H}(c + di) = ac - bd+(ad + be)i. 

The quotient obtained by the division of a + bi by c + di is 
defined to be a third imaginary symbol x + yi, such that when it is 
multiplied by c + di, the product is a + bi The equality 

a -f bi = (c + di) (x + yf) 

is equivalent, according to the rules of multiplication, to the two 
relations cx 

whence we obtain 




4 ELEMENTS OF THE THEORY [I, 1 

The quotient obtained by the division of a -j- In by c + di is repre- 
sented by the usual notation for fi actions in algebra, thus, 

a + bi 



A convenient way of calculating x and ?/ is to multiply numerator 
and denominator of the fraction by c d i and to develop the 
indicated products 

All the properties of the fundamental operations of algebra can be 
shown to apply to the operations carried out on these imaginary sym- 
bols Thus, if A, B, C, denote complex numbers, we shall have 

,4 B=B A, A B C=J (B C), A (B + C) = AB + AC, 

and so on. The two complex quantities a + bi and a bi aie said 
to be conjugate imaginanes The two complex quantities a + bi and 
a bi, whose sum is zeio, aie said to be negatives of each other 
or symmetric to each othei 

Given the usual system of lectangular axes in a plane, the complex 
quantity a + bi is lepresented by the point M of the plane xOy, whose 
cooidmates are x = a and y = b In this way a concrete representa- 
tion is given to these puiely symbolic expressions, and to eveiy 
proposition established for complex quantities there is a correspond- 
ing theorem of plane geometry But the greatest advantages resulting 
from this representation will appear later Real numbers correspond 
to points on the ct-axis, which for this reason is also called the aaib 
of reals Two conjugate imagmaries a + bi and a II correspond to 
two points symmetrically situated with respect to the o?-axis Two 
quantities a + bi and a bl are lepiesented by a pair of points 
symmetric with respect to the ongin The quantity a + In, which 
corresponds to the point M with the coordinates (a, ft), is sometimes 
called its affix.* When there is no danger of ambiguity, we shall 
denote by the same letter a complex quantity and the point which 
represents it 

Let us ;jom the ongin to the point M with coordinates (a, b) by a 
segment of a straight line The distance OM is called the absolute 
value of a + bij and the angle through which a ray must be turned 
from Ox to bring it in coincidence with OM (the angle being measured, 
as in trigonometry, from Ox toward Oy) is called the angle of a + bi 

* This term is not much used m English, but the French frequently use the coire- 
sponding word affixe TRANS 



GENERAL PRINCIPLES 



Let p and. a> denote, respectively, the absolute value and the angle of 
a -f fa > between the real quantities <z, I, p, o> there exist the two rela- 
tions a = p cos a), Z> = p sin <o, whence we have 



cos <o = 



sin a? = 



The absolute value p, which is an essentially positive number, is 
determined without ambiguity , whereas the angle, being given only 
by means of its trigonometric functions, is deteimined except for an 
additive multiple of 2 TT, which was evident from the definition itself. 
Hence every complex quantity may have an infinite number of 
angles, forming an arithmetic progression in which the successive 
terms differ by 2 TT. In order that two complex quantities be equal, 
their absolute values must be equal, and moieovei their angles must 
differ only by a multiple of 2 TT, and these conditions are sufficient. 
The absolute value of a complex quantity z is represented by the 
same symbol \&\ which is used for the absolute value of a real 
Quantity 

Let # = a + bi, &' = a* -\- b'i be two complex numbers and rn, m' 
the corresponding points , the sum & -f ' is then represented by the 
point m", the vertex of the parallelogram constructed upon Om, Om f . 
The three sides of the tuangle Om m n 
(Fig 1) are equal respectively to the 
absolute values of the quantities , z 1 , 
& -f- 2' From this we conclude that the 
absolute value of the sum of two quanti- 
ties u less than or at most equal to the 

sum of the absolute values of the two 

quantities, and greater than or at least 
equal to their difference Since two 
quantities that are negatives of each, 
other have the same absolute value, the theorem is also true for 
the absolute value of a difference Finally, we see in the same way 
that the absolute value of the sum of any number of complex 
quantities is at most equal to the sum of their absolute values, the 
equality holding only when all the points representing the different 
quantities are on the same ray starting from the origin. 

If through the point m we draw the two straight lines mx r and 
my' parallel to Ox and to Oy, the coordinates of the point m' in this 
system of axes will be a' a and b' b (Fig 2). The point m 1 
then represents # f # in the new system , the absolute value of 




x 



FIG. 1 



y 




6 ELEMENTS OF THE THEORY [I, 1 

#' & is equal to the length mm', and the angle of ' 2 is equal to 
the angle which the direction mm' makes with mx' Draw thiough 

a segment Oi^ equal and par- 
allel to mm' , the extremity m 1 of 
this segment represents f & in 
the system of axes Ox, Oy But 
the figuie Oni'vi^ is a parallelo- 
gram , the point m 1 is therefore 
the symmetric point to m with 
respect to c, the middle point 
of Om' 

_!_ J.VT 4 

Finally, let us obtain the for- 
mula which gives the absolute value and angle of the product of any 
number of factors Let 

z k = pi (cos <DJL 4~ * sin o> A ), (7c = 1, 2, , ri), 

be the factors , the rules for multiplication, together with the addi- 
tion f orrnulse of trigonometry, give for the product 

4- i sm (^ 4 <t> 2 + ' + 'Ol 

which shows that the absolute value of a product is equal to the 
product of the absolute values, and the angle of a product is equal to 
the siim of the angles of the factors From this follows very easily 
the well-known formula of Be Moivre 

cos m<t) 4- i sin w = (cos <o 4 * sin >) m } 

which contains in a very condensed form all the trigonometric for- 
mulse for the multiplication of angles 

The introduction of imaginary symbols has given complete gener- 
ality and symmetry to the theory of algebraic equations It was in 
the treatment of equations of only the second degree that such ex- 
pressions appeared for the first time Complex quantities are equally 
important in analysis, and we shall now state precisely what mean- 
ing is to be attached to the expression a function of a complex 
variable. 

2. Continuous functions of a complex variable. A complex quantity 
z =s x 4- yi") where x and y are two real and independent variables, 
is a complex variable If we give to the word function its most 
general meaning, it would be natural to say that every other complex 
quantity u whose value depends upon that of is a function of . 



I, 3] GENERAL PRINCIPLES 7 

Certain familiar definitions can be extended directly to tliese func- 
tions Thus, we shall say that a function u =f(z) is continuous if 
the absolute value of the difference f(& + &)/() approaches zeio 
when the absolute value of h approaches zero, that is, if to every 
positive number we can assign another positive number TJ such that 

|/(* + A) -/(*)!< 
provided that | Ji \ be less than 77 

Asenes > (*) + 1 (*)+ +.(*)+ , 

whose terms are functions of the complex vaiiable # is uniformly 
convergent in a region A of the plane if to eveiy positive number e 
we can assign a positive integer N such that 



for all the values of & in the region A, provided that n ^ N It 
can be shown as before (Vol I, 31, 2d ed , 173, 1st ed ) that if a 
series is uniformly convergent in a region A, and if each of its 
terms is a continuous function of 111 that region, its sum is itself 
a continuous function of the variable & in the same region. 

Again, a series is uniformly convergent if, for all the values of 
consideied, the absolute value of each term ^| 1S ^ ess tnan tne 
corresponding term ? H of a convergent series of real positive con- 
stants The series is then both absolutely and umfoimly convergent. 

Every continuous function of the complex variable is of the 
form u = P(x 9 2/) + Q(, y)i, where P and Q are real continuous 
functions of the two real variables oe, y~ If we were to impose no 
other restrictions, the study of functions of a complex variable 
would amount simply to a study of a pair of functions of two real 
variables, and the use of the symbol i would introduce only illusory 
simplifications In order to make the theory of functions of a com- 
plex vaiiable present some analogy with the theoiy of functions of a 
real variable, we shall adopt the methods of Cauchy to find the con- 
ditions which the functions P and Q, must satisfy in order that the 
expression P + Qi shall possess the fundamental properties of func- 
tions of a real variable to which the processes of the calculus apply 

3. Analytic functions. If /() is a function of a real variable x 
whach has a derivative, the quotient 



h 



8 ELEMENTS OF THE THEORY [I, 3 

approaches f(x) when 7i approaches zero. Let us determine in the 
same way under what conditions the quotient 

AM A/> + &AQ 



As: AT/ 4- 

will approach a definite limit when the absolute value of A# approaches 
zero, that is, when Aa? and Ay approach zero independently. It is 
easy to see that this will not be the case if the functions P(x } y) and 
Q(x, y) are any functions whatever, for the limit of the quotient 
Aw/A# depends in general on the latio Ay/Ax, that is, on the way 
in which the point repiesenting the value of -f- h approaches the 
point representing the value of z 

Let us first suppose y constant, and let us give to x a value x -f- A# 
differing but slightly from x , then 

Att_P(s + As,y)-.P(ar,y) | ^ Q (x + Ax, y) - Q (x, y) 

A# Ax Ace 

In order that this quotient have a limit, it is necessary that the 
functions P and Q possess partial derivatives with respect to x, and 

in that case Att 8/> 8fl 

lim = -M . 
As <?aj cos 

Next suppose x constant, and let us give to y the value y + A?/ , we 
have 



y + Ay) - P(x, y) ^ <a(ag, ?/ + A?/) - Q(g, y) 



Ay 
and in this case the quotient will have for its limit 

dQ .dP 

5 -- 1-%- 
dy dy 

if the functions P and Q possess partial derivatives with respect to y 
In order that the limit of the quotient be the same in the two cases, 
it is necessary that 

a/>_<2 ^!__Q 
^ ' dx dy dy"" dx 

Suppose that the functions P and Q satisfy these conditions, and 
that the partial derivatives dP/dx, dP/dy, dQ/dx, dQ/dy are con- 
tinuous functions. If we give to x and y any increments whatever, 
Aaj, Ay, we can write 

AP = P(x + Aa, y + &y)-P(x + bx,y) + P(x + Aa?, y)- P(x,y) 
(x + Ax, y + 0Ay) + Aa;P; (as + 0'Aa, y) 



I, 3] GENERAL PRINCIPLES 9 

where and 0' are positive numbers less than unity , and in the 
same way 

AQ = A[<(aj, y) + e'] + Ay [<&(, y) + <], 

where 6, e f , 1? cj approach zero with Ax and Ay The difference 
Aw = AP 4- *AQ can be written by means of the conditions (1) in 
the form, 



where 17 and 17' are infinitesimals We have, then, 
*!*=:! 4. dQ 



Ace 

If 1 17 1 and 1 77'! are smaller than a number a, the absolute value of the 
complementary term is less than 2 a This term will therefore ap- 
proach zero when Ax and Ay approach zero, and we shall have 

. A oP , .dQ 
lm "fe + ^- 

The conditions (1) are then necessary and sufficient in order that the 
quotient Aw/A# have a unique limit for each value of , provided that 
the partial derivatives of the functions P and Q be continuous The 
function u is then said to be an analytic function * of the variable z, 
and if we represent it by f(z), the derivative / f () is equal to any 
one of the following equivalent expressions : 

/ON ^/x dp , 3 30* - dp dp dp a 3. .dQ 
( 2 ) /W^T" + *T" "a -- *"fl" !B3 1 -- '^ = T~ + *^~" 
x x '.^ y &c ^cc ^y 3y ^a? oy oy ex 

It is important to notice that neither of the pair of functions 
P(x, y), Q(x, y) can be taken arbitrarily. In fact, if P and Q have 
derivatives of the second order, and if we differentiate the first of 
the relations (1) with lespect to x, and the second with respect to y, 
we have, adding the two resulting equations, 



* Cauchy made frequent use of the term monogene, the equivalent of which, mono- 
gentc, is sometimes used in English The term synectique is also sometimes used in 
French We shall use by preference the term analytic, and it will be shown latei 
that this definition agrees with the one which has already been given (I, 197, 
2d ed., 191, 1st ed) 



10 ELEMENTS OF THE THEORY [I, 3 

We can show in the same way that AQ = The two functions 
p(x, ?/), Q(x, y) must therefore be a pair of solutions of Laplace's 
equation. 

Conversely, any solution of Laplace's equation may be taken for 
one of the functions P or Q, For example, let P (x, ?/) be a solution 
of that equation , the two equations (1), where Q is regarded as an 
unknown function, aie compatible, and the expression 



Uc.rt 
.. 



which is determined except for an arbitrary constant C, is an analytic 
function whose real part is P(x, y) 

It follows that the study of analytic functions of a complex van- 
able 3 amounts essentially to the study of a pan ot functions 
P(OJ, ?/), Q(a-, y) of two real variables x and ?/ that satisfy the 
lelations (1) It would be possible to develop the whole theoiy with- 
out making use of the symbol i * 

We shall continue, however, to employ the notation of Cauchy, but 
it should be noticed that there is no essential difference between the 
two methods Every theorem established for an analytic function 
/(#) can be expressed immediately as an equivalent theorem relat- 
ing to the pair of functions P and (2, and conversely 

Examples The function u = x 2 y 2 + 2xyi is an analytic function, for it 
satisfies the equations (1), and its denvative is 2x + 2yz = 2z , m fact, the func- 
tion is simply (x + yi) z = z 2 On the other hand, the expression v x yi is not 
an analytic function, for we have 

Av __ Ax i Ay _ 

""" "~ 



Az Ax + lAy . , Ay 

1-f i 

Ax 

and it is obvious that the limit of the quotient Av/Az depends upon the limit of 
the quotient Ay /Ax 

If we put x == p cos w, y = p sin , and apply the formulae for the change of 
independent variables (I, 63, 2d cd , 38, 1st ed , Ex II), the relations 
(1) become 
/QA $P dQ <5Q dP 

(9) = p 1 = p > 

and the derivative takes the form 



dp 

* This is the point of view taken by the Geiman mathematicians who follow 
Eiemann 



I, 4] GENERAL PRINCIPLES 11 

It is easily seen on applying these formulae that the function 

jpn = pm ( cos mtl} 4. i gin mw ) 

is an analytic function of z whose derivative is equal to 

s w i sin w) = 



4. Functions analytic throughout a region. The preceding general 
statements are still somewhat vague, foi so far nothing has been 
said about the limits between which z may vary. 

A portion A of the plane is said to be connected, or to consist of 
a single piece, when it is possible to join any two points whatever 
of that portion by a continuous path which lies entirely in that 
portion of the plane. A connected portion situated entirely at a 
finite distance can be bounded by one or several closed curves, 
among which there is always one closed curve which forms the 
exterior boundary A portion of the plane extending to infinity may 
be composed of all the points exterior to one or moie closed curves ; 
it may also be limited by curves having infinite branches We shall 
employ the term region to denote a connected portion of the plane 

A function f(&) of the complex variable a? is said to be analytic * 
in a connected region A of the plane if it satisfies the following 
conditions 

1) To every point & of A corresponds a definite value of /(#) , 

2) /(#) is a continuous function of # when the point # varies in 
A, that is, when the absolute value of f(z + /0""/W approaches 
zero with the absolute value of h , 

3) At every point & of A, f(z) has a uniquely determined deriva- 
tive /'() , that is, to every point corresponds a complex number 
/ r () such that the absolute value of the difference 



n/ 

approaches zero when \h\ approaches zero Given any positive num- 
ber c, another positive number t\ can be found such that 



(4) i/^ + fc)-/^) 

if | h\ is less than ^ 

For the moment we shall not make any hypothesis as to the values 
of /(#) on the curves which limit A. When we say that a function 
is analytic in the interior of a region A bounded by a closed curve T 

* The adjective hotomorphic is also often used. TBAKS 



12 ELEMENTS OF THE THEORY [I, 4 

and on the boundary curve faelf, we shall mean by this that f(z) is 
analytic in a region Jl containing the boundary cmve T and the 
region A 

A function /() need not necessaiily be analytic throughout its 
region of existence It may have, in general, singular points, which 
may be of veiy varied types It would be out of place at this point 
to make a classification of these singular points, the veiy nature of 
which will appear as we proceed with the study of functions which 
we are now commencing 

5. Rational functions. Since the rules which give the derivative of 
a sum, of a product, and of a quotient are logical consequences of the 
definition of a derivative, they apply also to functions of a complex 
variable The same is true of the vule for the derivative of a func- 
tion of a function Let u =sf(Z) be an analytic function of the 
complex variable Z , if we substitute for Z another analytic function 
< (#) of another complex varrable &, u is still an analytic function of 
the variable 2. We have, in fact, 

Aw _ A?t ^ z . 
A* ~~ &Z A* ' 

when |A&| approaches zero, |AJ| approaches zero, and each of the 
quotients AW/A-2T, A-^T/As approaches a definite limit Therefore the 
quotient A-w/A itself approaches a limit 



We have already seen (3) that the function 



is an analytic function of #, and that its derivative is mz m ~ l . This 
can be shown directly as in the case of real variables In fact, the 
binomial formula, which results simply from the properties of multi- 
plication, obviously can be extended in the same way to complex 
quantities Therefore we can write 



1 1 A 

where m is a positive integer ; and from, this follows 



I, 6] 



GENERAL PRINCIPLES 



13 



It is cleat that the right-hand side has ma**- 1 for its limit when the 
absolute value of h appioaches zero. 

It follows that any polynomial with constant coefficients is an 
analytic function thioughout the whole plane A rational function 
(that is, the quotient of two polynomials P(, Q(z), which we may 
as weU suppose prime to each othei) is also in general an analytic 
function, but it has a certain number of singular points, the roots of 
the equation Q(z)= It is analytic in every region of the plane 
which does not include any of these points 

6. Certain irrational functions. When a point & descubes a continu- 
ous curve, the coordinates x and y, as well as the absolute value p, 
vary in a continuous manner, and the same is also tiue of the angle, 




x 



FIG 30 




36 



provided the curve described does not pass through the origin. If 
the point & describes a closed curve, #, y, and p return to their 
original values, but for the angle o> this is not always the case. If 
the origin is outside the region inclosed by the closed curve (Eig. 3 &), 
it is evident that the angle will return to its original value , but this 
is no longer the case if the point & describes a curve such as M^NPM^ 
or MftpqM^ (Fig 3 ) In the first case the angle takes on its original 
value increased by 2 TT, and in the second case it takes on its original 
value increased by 4 TT. It is clear that # can be made to describe 
closed curves such that, if we follow the continuous variation of the 
angle along any one of them, the final value assumed by <D will differ 
from the initial value by 2 mr, where n is an arbitrary integer, posi- 
tive or negative. In general, when describes a closed curve, the 



14 ELEMENTS OF THE TIIEOKY [I, 6 

angle of 2 a returns to its initial value if the point a lies outside 
of the region bounded by that closed cuive, but the cuive described 
by & can always be chosen so that the final value assumed by the 
angle of & a will be equal to the initial value increased by 2n7r 
Let us now consider the equation 

(5) f = , 

where m is a positive integer To every value of 2, except 2 = 0, 
there are m distinct values of u which satisfy this equation and 
therefore correspond to the given value of z In fact, if we put 

& = p (cos <o + *> sin <o), u = r (cos < + I sm <), 
the relation (5) becomes equivalent to the following pan 

w<}> = o) + 27i TT 



From the first we have i = p l/wl , which means that > is the wth anth- 
metic root of the positive imnibei p , from the second we have 



To obtain all the distinct values of we Lave only to givo to the 
arbitrary integer 7c the m consecutive integral values 0, 1, 2, , m 1 , 
in this way we obtain expiessions for the w, roots of the equation (5) 

/^ ^f /<o + 27r\, . /W + 2&7TY1 

(6) Wfc = ^[e OS (__^ , 

(* = 0,1, 2, ,w-l). 

It is usual to represent by 1M any one of these roots 

Wlien the variable z describes a continuous curve, each of these 
roots itself varies in a continuous manner If s describes a closed 
curve to which the oxigin is exteiior, the angle w comes back to its 
original value, and each of the roots U Q , u v , u m ^ describes a 
closed curve (Fig 4 a) But if the point z describes the curve 
JM Q NPM Q (Fig 3 5), GJ changes to <w + 2 TT, and the final value of the root 
u % is equal to the initial value of the root w, +1 Hence the arcs 
described by the different roots form a single closed curve (Fig 4 5) 
These m roots therefore undergo a cyclic permutation when the 
variable describes m the positive direction any closed curve with- 
out double points that incloses the origin. It is clear that by making 
* describe a suitable closed path, any one of the roots, starting from 
the initial value u^ for .example, can be made to take on for its final 
value the value of any of the other roots. If we wish to maintain 
continuity, we must then consider these m roots of the equation (5) 



I, 6] 



GENERAL PRINCIPLES 



15 



not as so many distinct functions of *, but a,s m distinct branches of the 
same function The point = 0, about which the permutation of the 
m values of u takes place, is called a critical point or a Iranc7i point 




46 



In order to consider the m values of u as distinct functions of &, 
it will be necessaiy to disiupt the continuity of these roots along a 
line proceeding from the origin to infinity. We can represent this 
bieak m the continuity very concretely as follows imagine that m 
the plane of 2, which we may legard as a thin sheet, a cut is made 
along a ray extending from the ongin to infinity, for example, along 
the ray OL (Fig. 5), and that then the two edges of the cut are 
slightly separated so that there is no path along which the variable 
& can move dnectly fiom one edge to the other Under these circum- 
stances no closed path whatever can inclose the origin, hence to 
each value of z corresponds a completely detexnuned value u % of the 
m roots, which we can obtain by tak- 
ing for the angle <> the value included 
between a and a 2 TT But it must 
be noticed that the values of u t at two 
points m, m f on opposite sides of the 
cut do not approach the same limit as 
the points approach the same point of 
the cut The limit of the value of u^ 
at the point m 1 is equal to the limit of 
the value of ^ 8 at the point m, multi- 
plied by [cos (2 vr/m) + 1 sin (2 7r/m)] 

Each of the roots of the equation (5) is an analytic function. Let 
u be one of the roots corresponding to a given value & Q , to a value 
of near # Q corresponds a value of u nea/i u Instead of trying to 




16 ELEMENTS OF THE THEORY [I, 6 

find the limit of the quotient (u u^/(z s ), we can determine the 

limit of its reciprocal 

z s u m u% 



and that limit is equal to mu*-\ We have, then, for the denvative 



m u m ~ l m z 

or, using negative exponents, 

, 1 -1 

u 1 = 2 m 

w 

In order to be sure of having the value of the denvative which coi re- 
sponds to the root considered, it is bettei to make use of the expies- 
sion (l/w)(i05). 

In the interior of a closed curve not containing the origin each 
of the determinations of V& is an analytic function The equation 
u m = A (2 a) has also m roots, which permute themselves cyclically 
about the critical point 2 = a 

Let us consider now the equation 

(7) u* = A(z- e,)(* - 2 ) - (* - O, 

where e v e# , e n are n distinct quantities We shall denote by 
the same letters the points which represent these n quantities Let 

us set A = R (cos a + t sin a), 

e k = p k (cos <o L + i sin w^.), (>fe = 1, 2, , TI), 
^ = r(cos + i sin 0), 

where o) ft represents the angle which the straight-line segment e k z 
makes with the direction Ox. From the equation (7) it follows that 



Pn , = -a) 1 . <^ 

hence this equation has two roots that are the negatives of each other, 



(8) 



N*P 

/>)*[ 



+ >i + + u. + 2 

cos - " 



I, 7] GENERAL PRINCIPLES 17 

When the variable z describes a closed curve C containing within 
it p of the points e v e^ , e n , p of the angles <a v <*># , o n will 
increase by 27r, the angle of u^ and that of w 2 will therefoie in- 
crease by 2 J7r If P 1S even, the two loots leturn to their initial 
values , but if p is odd, they aie peimuted In particular, if the 
curve incloses a single point e t , the two loots will be permuted. The 
n points e t are branch points In order that the two roots x and u 2 
shall be functions of z that are always uniquely determined, it will 
suffice to make a system of cuts such that any closed curve whatever 
will always contain an even number of critical points. We might, 
for example, make cuts along lays proceeding from each of the 
points e % to infinity and not cutting each other But there are many 
other possible arrangements If, for example, there are four criti- 
cal points e v e 2 , 8 , e^ a cut could be made along the segment of a 
straight line e^ and a second along the segment e^e. 

7. Single-valued and multiple-valued functions. The simple exam- 
ples which we have ]ust treated bring to light a very important fact. 
The value of a function f(z) of the variable & does not always depend 
entirely upon the value of z alone, but it may also depend in a cer- 
tain measure upon the succession of values assumed by the variable 
z in passing from the initial value to the actual value in question, 
or, in other words, upon the path followed by the variable z. 

Let us return, for example, to the function u = V#. If we pass 
from the point Jf to the point M by the two paths MJKM and M Q PM 
(Fig. 3 #), starting in each case with the same initial value for u, we 
shall not obtain at M the same value for u, for the two values 
obtained for the angle of z will differ by 2 TT. We are thus led to 
introduce a new distinction. 

An analytic f unction f(z) is said to be singles alued* in a legion 
A when all the paths in A which go from a point z^ to any other point 
whatever z lead to the same final value for /(#). When, however, 
the final value of f(z) is not the same for all possible paths in -4, 
the function is said to be 'multiple-valued t A function that is 
analytic at every point of a region A is necessarily single-valued in 
that region. In general, in order that a function f(z) be single- 
valued in a given region, it is necessary and sufficient that the func- 
tion return to its original value when the variable makes a circuit of 



* In French the term umforme or the term rnonodrome is used TRANS 
t In French the term multiforme is used TEANS 




18 ELEMENTS OF THE THEORY [l,7 

any closed path whatevei If, in fact, m going fiom the point A to 
the point B by the two paths AMB (Fig 6) and ANB, we airive in 
the two cases at the point B with the same determination of f(z), it 
is obvious that, when the vanable is made to describe the closed 
cuive A MBNA, we shall return to the point 
A with the initial value of /() 

Conversely, let us suppose that, the vanar 
ble having descubed the path AMBNA, we 
return to the point of departure with the 
initial value ^ ; and let it^ be the value of the 

function at the point B after z has described the path AMB. When 
z deseiibes the path BNA, the function starts with the value u t and 
amves at the value ^ ; then, conversely, the path ANB will lead 
from the value U Q to the value u^ that is, to the same value as the 
path AMB 

It should be noticed that a function which is not single-valued in a 
legion may yet have no critical points in that region Consider, for 
example, the poition of the plane included between two concentric cir- 
cles (\ r" having the origin for center. The function u = r- v in has no 
cutical point in that region , still it is not single-valued in that region, 
for if z is made to describe a concentric circle between C and 6 y ', the 
function g 1/m will be multiplied by cos (2 TT/WI) -f i sin (2 ir/m) 



JL POWER SERIES WITH COMPLEX TERMS ELEMENTARY 
TRANSCENDENTAL FUNCTIONS 

8. Circle of convergence. The reasoning employed in the study of 
power series (Vol I, Chap IX) will apply to power series with 
complex terms ; we have only to replace in the reasoning the phrase 
" absolute value of a real quantity" by the corresponding one, 
" absolute value of a complex quantity " We shall recall briefly the 
theorems and results stated there Let 

(9) + V + a^ + + <v~ n + 

be a power series in which the coefficients and the variable may have 
any imaginary values whatevei Let us also consider the series of 
absolute values. 



(10) AQ+A^+A^-I \~A n i+ , 

where ^ = [aj, r- = |#] We can prove (I, 181, 2d ed ; 177, 
1st ed ) the existence of a positive number 12 such that the series 



I, 8] POWER SERIES WITH COMPLEX TERMS 19 

(10) is convergent foi eveiy value of )><R, and diveigent for every 
value of r>R The number R is equal to the leciprocai of the 
greatest limit of the terms of the sequence 



and, as particular cases, it may be zero or infinite 

Prom these pioperties of the number R it follows at once that the 
series (9) is absolutely convergent when the absolute value of z is 
less than R It cannot be convergent for a value Z Q of & whose abso- 
lute value is gieater than R, for the series of absolute values (10) 
would then be convergent for values of r greater than R (I, 181, 
2d ed , 177, 1st ed ) If, with the origin as center, we descube in 
the plane of the variable z a circle C of ladius R (Fig. 7), the power 
seiies (9) is absolutely convergent for every value of z inside the 
circle (7, and divergent for every value of z outside , for this reason 
the circle is called the circle of convergence In a point of the circle 
itself the series may be convergent or divergent, according to the 
particular series * 

In the intenoi of a circle <?' concentric with the first, and with a 
ladius R 1 less than 72, the series (9) is uniformly convergent For 
at every point within C 1 we have evidently 



and it is possible to choose the integer n so large that the second 
member will be less than any given positive number e, whatever %> 
may be From this we conclude that the sum of the series (9) is a 
continuous function /() of the variable % at every point within the 
circle of convergence ( 2) 

By differentiating the seiies (9) repeatedly, we obtain an unlimited 
number of power series, /,(*), /,(*), . , /;(*), . , which have the 
same circle of convergence as the first (I, 183, 2d ed. ; 179, 
1st ed ) We prove in the same way as in 184, 2d ed , that f^z) 
is the denvative of /(), and in general that f n (z) is the derivative 

* Let/(z) = SanZ* 1 be a power series whose radius of convergence JR is equal to 1 
If the coefficients a , a^ a s , , are positive decreasing numbeis such that a n ap- 
proaches zeio when n inci eases indefinitely, the series is convergent in every point 
of the circle of convergence, except perhaps for z= 1 In fact, the seiies Sz, where 
1 2 1 = 1, is indeterminate except for zi, foi the absolute value of the sum of the fiist 
n terms is less than 2/| 1 - z | , it will suffice, then, to apply the reasoning of 166, Vol I, 
based on the generalized lemma of Abel In the same way the sei les a - % z + a 2 z 2 - 
which is obtained fiom the piecedmg by replacing z by - z, is convergent at all the 
points of the circle 1*1-1, except perhaps f or z = - 1 (Cf I, 166 ) 



20 



ELEMENTS OF THE THEORY 



of f n _i(z) JSvery power series represents therefore an analytic func- 
tion in the interior of its circle of convenience There is an infinite 

sequence of denvatives of the 
given function, and all of them 
aie analytic functions in the 
same ciicle Given a point # 
inside the circle C, let us 
draw a circle c tangent to 
the circle C in the interior, 
with the given point as cen- 
ter, and then let us take a 
point a + A inside c , if r and 
p are the absolute values of 
and h, we have r + p<R 
(Fig. 7). The sum /( 4- A) 
of the series is equal to the 
sum of the double series 




FIG. 7 



+ a 



n (n 1) 
1 2 



when we sum by columns But this series is absolutely convergent, 
for if we replace each term by its absolute value, we shall have a 
double series of positive terms whose sum is 

A i + A 1 (r + p) + .+ A n (r + ,) + .... 

We can therefore sum the double series (11) by rows, and we have 
then, for every point # + h inside the circle c } the relation 



(12) 



The series of the second member is surely convergent so long as 
the absolute value of h is less than R r, but it may be convergent 
in a larger circle Since the functions / a (), / 2 ()j > /()? are 
equal to the successive derivatives of /(*), the formula (12) is 
identical with the Taylor development. 

If the series (9) is convergent at a point Z of the circle of con- 
vergence, the sum/() of the series is the limit approached by the 
sum /(#) when the point approaches the point Z along a radius 



I, 9] POWER SERIES WITH COMPLEX TERMS 21 

which, terminates in that point. We prove this just as in Volume I 
( 182, 2d ed , 178, 1st ed ), by putting * == 6Z and letting in- 
crease from to 1 The theorem is still true when z, remaining inside 
the circle, approaches Z along a curve which is not tangent at Z to 
the cncle of convergence * 

When the radius R is infinite, the circle of convergence includes 
the whole plane, and the function f(&) is analytic for every value 
of z We say that this is an integral function , the study of tran- 
scendental functions of this kind is one of the most important 
objects of Analysis t We shall study 111 the following paragraphs 
the classic elementary transcendental functions 

9. Double series Given a power series (9) with any coefficients whatever, we 
shall say again that a second power series ar B z re . whose coefficients are all real 
and positive, dominates the first series if f 01 every value of n we have | a n \ =i cc n . 
All the consequences deduced by means of dominant functions (I, 186-189, 
2d ed , 181-184, 1st ed ) follow without modification in the case of complex 
variables We shall now give another application of this theory 

Let 



(13) / () + f l (z) +/,() + + f n (z) + . . 

be a series of which each term is itself the sum of a power series that converges 
in a cncle of radius equal to or gi eater than the number E > 0, 



Suppose each term of the series (13) replaced by its development according to 
powers of z , we obtain thus a double series in which each column is formed by 
the development of a f unction f t (z) When that series is absolutely convergent 
for a value of z of absolute value />, that is, when the double series 



is convergent, we can sum the first double series by rows for every value of z 
whose absolute value does not exceed p. We obtain thus the development of 
the sum F(z) of the series (13) in powers of z, 



- + a m +..., (n = 0, 1, 2, ) 

This proof is essentially the same as that for the development of f(z -f h) in 
powers of h 

Suppose, for example, that the series f t (z) has a dominant function of the 
form M t r/(r z), and that the series SJf z is itself convergent In the double 

* See PICABD, Traite d'Analyse, Vol. II, p 73 

t The class of integral functions includes polynomials as a special case If there 
are an infinite number of terms in the development, we shall use the expression 
integral transcendental function TRANS 



22 ELEMENTS OP THE THEORY [i, 9 

series the absolute value of the general term is less than M t \z\ n /i" If | z | < r, 
the series is absolutely convergent, foi the senes of the absolute values is 
convergent and its sum is less than r'ZM l /(r | z |) 

10. Development of an infinite product m power series. Let 



be an infinite product wheie each of the functions u % is a continuous function 
of the complex variable z in the region D If the series SlT t , where Z7 t = | u, |, 
is uniformly conveigent in the region, F(z) is equal to the sum of a senes that 
is uniformly convergent in Z), and therefoie repiesents a continuous function 
(I, 175, 176, 2d ed.). When the functions u l aie analytic functions of 2, it fol- 
lows, fiom a general theorem which will be demonstrated latei ( 39), that the 
same is true of F(z) 

JTor example, the infinite product 



represents a function of z analytic thioughout the entire plane, for the series 
S|s| 2 /w 2 is uniformly convergent within any closed curve whatevei This 
product is zero for z = 0, 1, 2, and for these values only. 

We can prove directly that the product F(z) can be developed m a power 
senes when each of the functions u % can be developed m a power senes 



such that the double senes 



is convergent for a suitably chosen positive value of r. 
Let us set, as in Volume I ( 174, 2d ed ), 



It is sufficient to show that the sum of the series 
(14) 



which is equal to the infinite product F(z), can be developed in a power series. 
Now, if we set 

< 
it is clear that the product 



is a dominant function for v n It is therefore possible to arrange the series (14) 
according to powers of z if the following auxiliary senes 

(15) J + i + +<+- 

can be so arranged 

If we develop each term of this last series in power series, we obtain a 
double series with positive coefficients, and it is sufficient for our purpose to 



I, 11] POWER SERIES WITH COMPLEX TERMS 23 

prove that the double series converges when z is replaced by r. Indicating by 
UK and V' n the values of the functions u n and i for z = r, we have 

and therefore 
or, again, 



When n increases indefinitely, the sum UQ + - + U^ approaches a limit, since 
the series SlT^ is supposed to be convergent The double senes (15) is then 
absolutely convergent if | z \ s r , the double senes obtained by the development 
of each term v n of the series (14) is then a f ortioii absolutely convergent within 
the circle C of radius r, and we can arrange it according to integral powers of z 
The coefficient b p of zP m the development of F(z) is equal, from the above, to the 
limit, as n becomes infinite, of the coefficient 6pof z* in the sum i? + v x + -f v n , 
or, what amounts to the same thing, in the development of the product 



Hence this coefficient can be obtained by applying to infinite products the 
ordinary rule which gives the coefficient of a power of z in the product of a 
finite number of polynomials For example, the infinite product 



can be developed according to powers of z if | z \ < 1 Any power of z whatever, 
say Z N , will appear in the development with the coefficient unity, f 01 any posi- 
tive integer N can be written in one and only one way in the form of a sum of 
powers of 2 We have, then, if | z \ < I , 



(16) 



1 z 
which can also be very easily obtained by means of the identity 



11. The exponential function. The arithmetic definition of the ex- 
ponential function evidently has no meaning when the exponent is 
a complex number. In order to generalize the definition, it will be 
necessary to start with some property which is adapted to an exten- 
sion to the case of the complex variable We shall start with the 
property expressed by the functional relation 



Let us consider the question of determining a power series /(), con- 
vergent in a circle of radius R, such that 

(17) /( + *')=/</(*') 

when, the absolute values of , ', a + s' are less than R, which will 



24 ELEMENTS OF THE THEORY [I, 11 

surely be the case if \\ and |'| are less than R/2 If we put 2' = 
in the above equation, it becomes 



Hence we must have /(O) = 1, and we shall write the desired series 



Let us replace successively in that series & by X, then by X', where 
X and X' are two constants and t an auxiliary variable ; and let us 
then multiply the resulting series This gives 



On the other hand, we have 



The equality /(Xi + X r O=/(XO/(A. f #) is to hold for all values of 
X, X', t such that |X| < 1, |X'| < 1, |*| < /2/2. The two series must 
then be identical, that is, we must have 




and from this we can deduce the equations 

a n = a n _ l a 1 , a w =a n 2 a 2 , , 
all of which can be expressed in the single condition 



where jp and gf are any two positive integers whatever. In order to 
find the general solution, let us suppose q = 1, and let us put 
successively p = 1, p = 2, JK> = 3, ; from this we find 2 = af , then 
<& 8 = 0,^= aj, , and finally a n = aj. The expressions thus obtained 
satisfy the condition (18), and the series sought is of the form 






I, 11] POWER SERIES WITH COMPLEX TERMS 25 

This series is convergent in the whole plane, and the relation 
/(* + *')=/(*) A* 1 ) 

is true for all values of & and ^ 

The above series depends upon an arbitrary constant a v Taking 
a l = 1, we shall set 



so that the geneial solution of the given problem is e a i z The inte- 
gral function e z coincides with the exponential function e x studied in 
algebra when is real, and it always satisfies the relation 



whatever z and z* may be The derivative of e z is equal to the func- 
tion itself. Since we may write by the addition formula 



in order to calculate e z when & has an imaginary value x + yi, it is 
sufficient to know how to calculate e vt Now the development of e m 
can be written, grouping together terms of the same kind, 



We recognize in the second member the developments of cos y and 
of siny, and consequently, if y is real, 



Replacing e^ by this expression in the preceding formula, we have 
(19) &* + Vl = e*(cos y + i sin y) , 

the function e*+w has e* for its absolute value and y for its angle. 
This formula makes evident an important property of e* , if z 
changes to & + 2 m, x is not changed while y is increased by 2 TT ? 
but these changes do not alter the value of the second member of 
the formula (19) We have, then, 



that is, the exponential function e* has the period 2 m 

Let us consider now the solution of the equation e* = A, where A 
is any complex quantity whatever different from zero Let p and <u 
be the absolute value and the angle of A , we have, then, 

ga+y* = a x (cos y + ^ sm y) = p (cos o> + i sin <o), 



26 ELEMENTS OF THE THEORY [I, ll 

from which it follows that 

e x = p, y = <o + 2 JCTT. 

From the first relation we find x = log p, where the abbreviation log 
shall always be used foi the natmal logarithm of a leal positive 
number. On the other hand, y is determined except for a multiple 
of 2 TT If A is zero, the equation & = leads to an impossibility 
Hence the equation e* = A, wJiere A is not zero, has an ^nfin^te num- 
ber of roots given ly the expression logp 4- i( + 27c7r), the equation 
Q Z = has no roots, real or imaginary 

Note We might also define e z as the limit approached by the poly- 
nomial (1 + &/m) m when m becomes infinite The method used in 
algebia to prove that the limit of this polynomial is the series e* can 
be used even when & is complex 

12. Trigonometric functions. In order to define sin & and cos # 
when 2 is complex, we shall extend directly to complex values the 
series established for these functions when the variable is real 
Thus we shall have 



(20) 






These are integral transcendental functions which have all the 
properties of the trigonometric functions. Thus we see from the 
formulae (20) that the derivative of sin # is cos #, that the derivative 
of cos & is sin #, and that sm % becomes sin , while cos % does 
not change at all when is changed to & 

These new transcendental functions can be brought into very close 
relation with the exponential function In fact, if we write the ex- 
pansion, of e* 1 , collecting separately the terms with and without the 
factor L 



21 ' 4! ' 
we find that that equality can be written, by (20), in the form 

&*> ss cos * + i sm . 
Changing to 2, we have again 

e~~** = cos r 
and from these two relations we derive 



I, 12] POWER SERIES WITH COMPLEX TERMS 27 



cos s = - - - > sins 



2 ' OA "~ 2i 

These are the well-known formulae of Euler which express the 
trigonometric functions in terms of the exponential function They 
show plainly the periodicity of these functions, for the right-hand 
sides do not change when we replace z by z 4- 2 TT Squaring and 
adding them, we have 

cos 2 z 4- sin 2 2 = 1 

Let us take again the addition formula e (z+ * 0t = e 2 *e a/l , or 

cos (z + z') 4- i sin (z 4- z') 

= (cos z 4- i sin z) (cos z' 4- i sin z 1 ) 

= cos 2 cos z 1 sin sin z' i-^ (sin 2 cos 2' + sin z 1 cos 2), 

and let us change z to 2, 2' to 2'. It then becomes 

cos (2 4- 2 1 ) * sin (2 4- 2*) 

= cos z cos 2' sin z sin ' t(sin 2 cos # f 4- sin z' cos 2), 

and from these two formulae we derive 

cos (z 4- #') =5 cos 2 cos z f sin 2 sin z' 
sin (2 4- 2') = sin z cos #' 4- sin z cos 2'. 

The addition formulae and therefore all their consequences apply for 
complex values of the independent variables Let us determine, for 
example, the real part and the coefficient of i in cos (x + yi) and 
sin (x + 2/1). We have first, by Euler's formulae, 



. e e , 

cos 7^ = - 5 - = coshy, 



5 - , = 

whence, by the addition formulae, 

cos (x + yi) = cos a; cos y sin oc sin yt = cos x cosh y t sin x sinh y, 
sin (x + y&) = sin x cos iy + os x sin 2/ = sin x cosh y + 1 cos x sinh T/ 

The other trigonometric functions can be expressed by means of 
the preceding For example, 

sin 2 10 st e~ s * 
tan 2 = - = T -TT-: - z ' 
cos 2 * 6**+ e"* 1 

which may be written in the form 



The right-hand side is a rational function of e 2 **; the penod of the 
tangent is therefore TT. 



28 ELEMENTS OF THE THEORY [I, 13 

13. Logarithms. Given, a complex quantity #, different fiom zero, 
we have already seen ( 11) that the equation e u = has an infinite 
numher of roots Let u = x + yi, and let p and u> denote the absolute 
value and angle of &, respectively Then we must have 

&* = p } f/ = 0) + 2 A 7T 

Any one of these roots is called the logarithm of and will be 
denoted by Log () We can write, then, 

Log (s) = log p + i (o> + 2 //TT), 

the symbol log being leserved for the ordinal y natuial, or Napierian, 
logarithm of a real positive number 

Every quantity, real or complex, different fiom zero, has an 
infinite number of logarithms, which form an anthmetic progies- 
sion whose consecutive terms diffei by 2 m In particular, if is a 
real positive number x, we have <o = Taking k = 0, we find again 
the ordinary logarithm , but there aie also an infinite number of 
complex values for the logarithm, of the form logos 4- 2 kiri If z is 
real and negative, we can take <o = TT , hence all the determinations 
of the logarithm are imaginary 

Let &' he another imaginary quantity with the absolute value />' 
and the angle o>'. We have 

Log (>') = log p' + 4 (' + 2 /C'TT) 
Adding the two logarithms, we obtain 

Log(s)+ Log(>')= logpp f 4- *|> + ' + 2(A + /C')TT] 

Since pp* is equal to the absolute value of ##', and o> + CD' is equal to 
its angle, this formula can be written in the form 

Log (*) + Log (V) = Log <X), 

which shows that, when we add any one whatever of the values of 
Log(#) to any one whatever of the values of Log(V), the sum Is one 
of the determinations of Log(^ f ). 

Let us suppose now that the variable s describes in its plane any 
continuous curve whatever not passing through the origin; along 
this curve p and <o vary continuously, and the same thing is true of 
the different determinations of the logarithm But two quite distinct 
cases may present themselves when the variable & traces a closed 
curve When starts from a point # and returns to that point after 
having described a closed curve not containing the origin within it, 
the angle o> of takes on again its original value <D O , and the different 



1, 13] POWER SERIES WITH COMPLEX TERMS 29 

determinations of the loganthm eoine back to their initial values. If 
we represent each value of the logarithm by a point, each of these 
points traces out a closed cuive On the contrary, if the vanable z 
describes a closed curve such as the curve M^NMP (Fig. 3 #), the 
angle increases by 27r, and each determination of the logarithm 
returns to its initial value increased by 2iri In general, when & 
describes any closed curve whatever, the final value of the logarithm 
is equal to its initial value increased by 2k7ri, wheie k denotes a 
positive or negative integer which gives the number of i evolutions 
and the direction through which the radius vector joining the origin 
to the point z has tinned. It is, then, impossible to considei the dif- 
ferent determinations of Log (z) as so many distinct functions of z 
if we do not place any restriction on the vanation of that vanable, 
since we can pass continuously from one to the other They aie so 
many branches of the same function, which aie permuted among 
themselves about the critical point z = 

In the interior of a region which is bounded by a single closed curve 
and which does not contain the origin, each of the determinations of 
Log (2) is a continuous single-valued function of z. To show that it 
is an analytic function it is sufficient to show that it possesses a 
unique derivative at each point Let z and z^ be two neighboring 
values of the variable, and Log(#), Log (24) the corresponding values 
of the chosen deteimmation of the logarithm When & l approaches 

2, the absolute value of Log ( a ) Log (z) approaches zero Let us put 
Log (*) = w, Log (^ = u^ , then 



When u^ approaches u, the quotient 

e tt i - e u 
u^ u 

approaches as its limit the derivative of e w , that is, e u or z. Hence 
the logarithm has a uniquely determined derivative at each point, 
and that derivative is equal to 1/2. In general, Log (2 a) has an 
infinite number of determinations which permute themselves about 
the critical point z = a, and its derivative is l/(s a) 

The function z m , where m is any number whatever, real or complex, 
is defined by means of the equality 



30 ELEMENTS OF THE THEORY [I, 13 

Unless m be a real lational number, this function possesses, just as 
does the logarithm, an infinite number of deteiminations, which per- 
mute themselves when the variable turns about the point & = It is 
sufficient to make an infinite cut along a ray fiom the origin in 
order to make each branch an analytic function in the whole plane 
The derivative is given by the expression 



and it is clear that we ought to take the same value for the angle 
of z in the function and m its derivative 

14. Inverse functions : arc sin z, arc tan z. The inverse functions 
of sin s, cos 2, tan & are defined in a similar way. Thus, the function 
u = arc sin & is defined by the equation 

2 = sin u 
In order to solve this equation for u, we write 

6 &~ m _ 2tu 1 
*~ 2^ ~T^' 

and we are led to an equation of the second degree, 

(22) 17 2 - 2^-1=0, 

to deteimine the auxiliary unknown quantity U = e ui . We obtain 
from this equation 

(23) U = fa Vl s a , 

or 

i 

(24) u = arc sm 2 = - Log (i& Vl # 3 ) 

The equation z = sin ^ has therefore two sequences of roots, which 
arise, on the one hand, from the two values of the ladical Vl # 2 , 
and, on the other hand, from the infinite number of determinations 
of the logarithm But if one of these determinations is known, 
all the others can easily be determined from it Let U 1 = p'e 11 *' and 
U" = p"e lw// be the two roots of the equation (22) ; between these 
two roots exists the relation 17 f i7"= 1, and therefore p'p" = 1, 
o)'-h o>" = (2 w + 1) TT. It is clear that we may suppose <o" = TT o> f , 
and we have then 



Log (U 1 ) = log P ' + <(' + 

Log (17") = - log P ' + j (w - o>' + 2 t"ir). 



I, 14] POWER SERIES WITH COMPLEX TERMS 31 

Hence all the determinations of arc sin 2 are given by the two 
formulae 

axe sins = CD' + 2 &'TT i log//, 

arc sing = TT + 2 &"TT <o' + 
and we may write 

(A) arc sin = u f + 

(B) arc sin* = (2 

where %' = a/ ^ logp'. 

When the variable describes a continuous curve, the various 
determinations of the logarithm in the formula (24) vary in general 
in a continuous manner. The only critical points that are possible 
are the points 2 = 1, around which the two values of the radical 
Vl g 2 are permuted, there cannot be a value of & that causes 
i Vl to vanish, for, if there were, on squaring the two sides 
of the equation iz = Vl # 2 we should obtain 1=0 

Let us suppose that two cuts are made along the axis of reals, one 
going from oo to the point 1, the other from the point + 1 to 
+ oo . If the path described by the variable is not allowed to cross 
these cuts, the different determinations of arc sins aie single-valued 
functions of 2. In fact, when the variable describes a closed curve 
not crossing any of these cuts, the two roots ?7 ; , U" of equation (22) 
also describe closed curves. None of these curves contains the 
origin in its interior. If, for example, the curve described by the 
root U 1 contained the origin in its interior, it would cut the axis Oy 
in a point above Ox at least once Corresponding to a value of U of 
the form ia(a > 0), the relation (22) determines a value (1 + cf)/2 a 
for s, and this value is real and > 1 The curve described by the 
point s would therefore have to cross the cut which goes from 
+ 1 to + oo 

The different determinations of arc sin z are, moreover, analytic 
functions of z * Eor let u and u be two neighboring values of 



* If we choose in C r =i2+Vl~2 2 the determination of the radical which reduces to 
1 when 2=0, the real part of U remains positive when the variable z does not cross 
the cuts, and we can put U= .Re**, where $ lies between --7T/2 and +<rr/2 The cor- 
responding value of (1/i) Log CT, namely, 

7 

is sometimes called the principal value of arc sm z It reduces to the ordinary deter- 
mination when z is real and hes between - 1 and + 1. 



32 ELEMENTS OF THE THEORY [I, i* 

arc sing, corresponding to two neighboring values z and ^ of the 
variable We have 

Uj U U l U 

#j # sin u^ sin u 

When the absolute value of it l u approaches zero, the preceding 
quotient has for its limit 

1 1 



cos u 



The two values of the derivative correspond to the two sequences 
of values (A) and (B) of arc sin & 

If we do not impose any lestnction on the variation of #, we can 
pass fiom a given initial value of arc sins to any one of the deter- 
minations whatever, by causing the variable & to describe a suitable 
closed curve In fact, we see fiist that when # descubes about the 
point = 1 a closed curve to which the point z = 1 is exterior, 
the two values of the radical Vl "* are pei muted and so we pass 
fiom a determination of the sequence (A) to one of the sequence (B) 
Suppose next that we cause z to descube a circle of radius R (R > 1) 
about the origin as center, then each of the two points U\ U" describes 
a closed curve. To the point z= + R the equation (22) assigns two 
values of U, U' = m^ U" = ^|3, where a and ft are positive , to the 
point ^ = J2 there coriespond by means of the same equation the 
values U 1 = to;', U' 1 = ip 1 , where a' and {? are again, positive 
Hence the closed curves described by these two points U', T7" cut the 
axis Oy in two points, one above and the other below the point , 
each of the logarithms Log (7"'), Log(?7") increases or diminishes 
by 2?r4 

In the same way the function aic tan a is defined by means of 
the relation tan u = 2, or 

1 e* - 1 



whence we have 2ta __ 3 + iz 

6 - 



and consequently J 1 

J arc tan z = ^ 

This expression shows the two logarithmic critical points i of the 
function arc tan &. When the variable z passes around one of these 
points, Log [(i z)/(i + #)] increases or diminishes by 2 iri, and 
arc tan increases or diminishes by TT 



I, 15] POWER SERIES WITH COMPLEX TERMS 33 

15. Application to the integral calculus. The derivatives of the func- 
tions which we have just defined have the same form as when the 
variable is real Conveisely, the rules for finding primitive functions 
apply also to the elementaiy functions of complex variables Thus, 
denoting by ff(z)dz a function of the complex variable & whose 
derivative is /(s), we have 

Adz = _ A 1 

(2 a) m m 1 (2 a)" 1 " 1 ^ 

=A Log(s a). 



These two formulae enable us to find a primitive function of any 
rational function whatever, with real or imaginary coefficients, pro- 
vided the roots of the denominator aie known. Consider as a special 
case a rational function of the real variable x with real coefficients 
If the denominator has imaginary roots, they occur in conjugate 
pairs, and each root has the same multiplicity as its conjugate. 
Let a + fti and a fii, be two conjugate roots of multiplicity p In 
the decomposition into simple fractions, if we proceed with the 
imaginary roots just as with the real roots, the root a + fii will 
furnish a sum of simple fractions 

M l + N l i M 2 + N 2 i M p + N p l 

'' 



x-a-fti (x-a-pi)*' (x-a- fii)* 

and the root a fti will furnish a similar sum, but with numerators 
that are conjugates of the former ones. Combining in the primitive 
function the terms which come from the corresponding fractions, we 
shall have, if p >1, 



p - 1 (* - - fa)"- 1 (x-a 



and the numerator is evidently the sum of two conjugate imaginary 
polynomials. If p = 1, we have 



CM^ + N^ 

J x-a-fii 






L + JVjt) Log [(as -a) - #.] + (M t - Nj,) Log [(* - a) + i] 



34 ELEMENTS OF THE THEORY [I, 15 

If we replace the logarithms by their developed expressions, there 
remains on the right-hand side 



M l log [(a? - of + /3 2 ] + 2 ^ aic tan ^f-^ 

It suffices to replace 

B , TT , x a 

arc tan c by 77 arc tan - 
x a J 2 p 

in order to express the result in the form in which it is obtained 
when imaginary symbols are not used 
Again, consider the indefinite integral 

dx 



which has two essentially different forms, according to the sign of 
A The introduction of complex variables reduces the two forms to a 
single one In fact, if in the formula 



we change x to ^x, there results 
dx 1. 



and the right-hand side represents precisely arc sin a?. 

The introduction of imaginary symbols in the integral calculus 
enables us, then, to reduce one formula to another even when the 
relationship between them might not be at all apparent if we were 
to remain always in the domain of real numbers 

We shall give another example of the simplification which comes 
from the use of imagmaries If a and 1) are real, we have 

J X ~~ a + bi~ < 

Equating the real parts and the coefficients of i, we have at one stroke 
two integrals already calculated (I, 109, 2d ed. ; 119, 1st ed.): 



_ , tf* (a cos bx + b sin bx) 

e cos bx dx = ^ 5 -: L j 

a a + lr 



7 7 ^^sinftaj & cos foe) 
e sin bx dx = ii $ , / - 



I, 16] POWER SERIES WITH COMPLEX TERMS 85 

In the same way we may reduce the integrals 

/ x m e ax cog fa tf Xy I xm e ax g^ fa fa 

to the integral fx m e <a+bl "> x dx, which can be calculated by a succession 
of integrations by parts, where m is any integer. 

16 Decomposition of a rational function of smz and cos^ into 
simple elements. Given a rational function of sin z and cos 2, 
.F(siii, cos #), if in it we replace sins and cos 2 by their expiessions 
given by Euler's formula, it becomes a rational function R() of 
t = e zl This function R (), decomposed into simple elements, will be 
made up of an integral part and a sum of fractions coming from 
the roots of the denominator of R (t) If that denominator has the 
root t = 0, we shall combine with the integral part the fractions aris- 
ing from that root, which will give a polynomial or a rational function 
R l (j)~'SK m t' in , where the exponent m may have negative values. 

Let t = a be a root of the denominator different from zero. That 
root will give rise to a sum of simple fractions 



The root a not being zero, let a: be a root of the equation e* 1 = a, , 
then !/( a) can be expressed very simply by means of ctn [(# #) 
We have, in fact, 

. * or .< 
eta. ^ 

whence it follows that 

1 1 



Hence the rational fraction /(#) changes to a" polynomial of degree 
w in ctn [(s ) 



The successive powers of the cotangent up to the nth can be ex- 
pressed in turn in terms of its successive derivatives up to the 
(n l)thj we have first 

d ctn z 



dz sin 2 * 



l ctn 2 *, 



36 ELEMENTS OF THE THEORY [I, 16 

winch enables us to express ctn 2 s 111 terms of d(cknz)/dz, and it is 
easy to show, by mathematical induction, that if the law is ti ue up 
to ctn w #, it will also be true for ctn tt+1 # The preceding polynomial 
of degree n in ctn[( &)/2] will change to a linear expression in 
ctn[(s #)/2] and its derivatives, 



or 



d n ~ l 



Let us proceed in the same way with all the roots b, c, , / of the 
denominator of R () different from zero, and let us add the results 
obtained after having replaced t by e zl in 11$) The given rational 
function F(sms, cos 2) will be composed of two parts, 

(25) .F( sin a, cos a) = $ () + * () 

The function $(#), which corresponds to the integral part of a 
rational function of the vanable, is of the form 



(26) $ (*) = C + S (tf /tt cos MS + j8 OT sin mst) 9 

where 7?i is an integer not zero On the other hand, ^(), which cor- 
responds to the fractional part of a rational function, is an expression 
of the form 



It is the function ctn[( a)/2] which here plays the role of the 
simple element, just as the fraction !/( a) does for a rational 
function The result of this decomposition of /''(sin , cos ) is easily 
integrated , we have, in fact, 



(27) 

and the other terms are integrable at once In order that the primi- 
tive function may be periodic, it is necessary and sufficient that all 
the coefficients C, j4 v $ v be zero. 

In practice it is not always necessary to go through all these suc- 
cessive transformations in order to put the function j?(sin #, cos ) into 
its final form (25) Let # be a value of * which makes the function 
F infinite We can always calculate, by a simple division, the 



I, 16] POWER SERIES WITH COMPLEX TERMS 37 

coefficients of (z a)" 1 , (z a)~ 2 , , in the part that is infinite 
for z = a (I, 188, 2d ed , 183, 1st ed ) On the other hand, we have 



where P (z #) is a power series , equating the coefficients of the 
successive poweis of (z a)- 1 in the two sides of the equation (25), 
we shall then obtain easily jj^ X, - -, Jl n 

Consider, for example, the function l/(cos # cos a) Setting 
e zl = t, e al = a, it takes the foim 

2 at 



The denominator has the two simple loots t = a, * = I/a, and the 
numerator is of lower degree than the denominator. We shall have, 
then, a decomposition of the form 



cos z cos a 

In order to determine jl, let us multiply the two sides by z a;, and 
let us then put z = a. This gives 1 /T = l/(2 sin a). In a similar 
manner, we find $ = V(2 sin <*) Replacing ^ and $ by these values 
and setting z = 0, it is seen that C = 0, and the formula takes the form 

1 = 1 

cos z cos a ~" 2 sin a 

Let us now apply the general method to the integral powers of sin z and 
of cos 2 We have, for example, 



Combining the terms at equal distances from the extremities of the expansion 
of the numerator, and then applying Euler's f ormulse, we find at once 

(2 cosz) = 2 cosmz + 2m cos(m 2)2 + 2 m \ "" ^cos(m 4)z + 

1 2 

If m is odd, the last term contains cos z , if m is even, the term which ends the 
expansion is independent of z and is equal to m '/ [(m/2) '] 2 . In the same way, 
if m is odd, 

(2 1 smz)* = 2 * sin mz 2 ww- sm(m 2)2 + 2z m ^" ^sin (m 4)2 - - - , 

and if m is even, m 

- ' 
; 2mcos(m 2)2+ + ( I) 2 



These formulae show at once that the primitive functions of (sm2) m and of 
(cos z) m are periodic functions of z when m is odd, and only then 



38 ELEMENTS OF THE THEORY [I, 16 

Note When the function F(8inz, eos#) has the penod IT, we can 
express it rationally in terms of &*** and can take foi the simple 
elements ctn (0 a), ctii(# /3), * 

17. Expansion of Log (1 + *). The transcendental functions which 
we have defined are of two kinds . those which, like e*j sin , cos *, are 
analytic in the whole plane, and those which, like Log &, arc tan ,, 
have singular points and cannot be represented by developments in 
power series convergent m the whole plane Nevertheless, such 
functions may have developments holding for certain paits of the 
plane We shall now show this for the logarithmic function 

Simple division leads to the elementary formula 



and if |*| <1, the remainder z*+ l /(l+z) approaches zero when n 
increases indefinitely Hence, m the interior of a circle C of radius 1 
we have -i 



Let F(&) be the senes obtained by integrating this series term by term 

-, ^2 ^8 *4 .tt+l 

^)-I-5 + F-I-- +<- 1 )%7+T + " ; 

this new series is convergent inside the unit circle and represents 
an analytic function whose derivative F'(z) is 1/(1 + z) We know, 
however, a function which has the same derivative, Log (1 + 0) It 
follows that the difference Log (1 -f 0) F() reduces to a constant.* 
In order to determine this constant it will be necessary to fix pre- 
cisely the determination chosen for the logarithm. If we take the 
one which becomes zero for = 0, we have for every point inside 

(28) Log(l + ) = |-| + |'-| + .... 

Let us join the point A to the point M, which represents (Fig. 8) 
The absolute value of 1 + * is represented by the length r = AM. 
Por the angle of 1 + z we can take the angle a which AM makes 
with AO, an angle which lies between 7r/2 and + Tr/2 as long as 
the point M remains inside the circle (7. That determination of the 



* In order that the denvative of an analytic function JT+ Yi "be zero, it is neces- 
sary that we haye (3) 0J5r/&e0, SF/Saj0, and consequently 
Xand Tare therefore constants. 



I, 17] 



POWER SERIES WITH COMPLEX TERMS 



39 



logarithm which becomes zero for & = is log r -f- ice ; hence the 
formula (28) is not ambiguous 




FIG 8 



Changing 2 to s in this formula and then subtracting the two 
expressions, we obtain 



If we now replace z by iz } we shall obtain again the development of 
arc tan # 



The series (28) remains convergent at every point on the circle of convergence 
except the point A (footnote, p. 19), and consequently the two series 

cos20 , cos 30 cos40 f 



sm 20 sm 3 sm 4 
._ (._ _+ ... 

are both convergent except f or = (2 k + 1) ir (cf I, 166) By Abel's theorem 
the sum of the series at M' is the limit approached by the sum of the series at 
a point M as M approaches M' along the radius OJkP If we suppose always 
between TT and + TT, the angle a will have for its limit 0/2, and the absolute 
value AM will have for its limit 2 cos (0/2) We can therefore write 

/ ft 0\ /, cos20 cos30 cos40 , 
log (2cos-)=cos0 _ + _+ , 

\ ] ft O * 





If In the last formula we replace by B IT, we obtain again a formula pre- 
viously established (I, 204, 2d ed , 198, 1st ed.) 



40 ELEMENTS OF THE THEORY [I, is 

18. Extension of the binomial formula. In a fundamental paper on 
power series, Abel set for himself the problem of determining the 
sum of the convergent series 



H , m , m ( m 
= 



m(m~l) 



2 i 
'* + 



for all the values of m and #, real or imaginary, provided we 
have | | < 1 We might accomplish this by means of a differential 
equation, in the manner indicated in the case of leal variables 
(I, 183, 2d ed , 179, 1st ed ) The following method, which gives 
an application of 11, is more closely related to the method fol- 
lowed by Abel. We shall suppose a fixed and \\ < 1, a&d we 
shall study the properties of </> (w, 2) considered as a function of m. 
If m is a positive integer, the function evidently reduces to the 
polynomial (1 + z) m . If m and m' are any two values whatever of 
the parameter m, we have always 

(30) < (w, *) < (m f , ) = < (m -f m', z) 

In fact, let us multiply the two series < (m, #), <j> (w', ) by the ordi- 
nary rule The coefficient of # p in the product is equal to 

(31) m p + m^^m{ -f m p _ 2 ^2 + + ?Vi-i + K>> 
where we have set for abbreviation 

m (m 1) - (m k + 1) 

m * = ^ - -. - *. 

The proposed functional relation will be established if we show 
that the expression (31) is identical with the coefficient of # p in 
<f> (m + m f , #), that is, witK (m + m')^ We could easily verify directly 
the identity 

(32) (m -f m') p ~ 



but the computation is unnecessary if we notice that the relation 
(30) is always satisfied whenever m and m' are positive integers 
The two sides of the equation (32) are polynomials m m and m' 
which are equal whenever m and m 1 are positive integers, they 
are therefore identical. 

On the other hand, <(#&, z) can be expanded in a power series 
of increasing powers of m. In fact, if we carry out the indicated 
products, < (m, &) can be considered as the sum. of a double series 



I, 18] POWER SERIES WITH COMPLEX TERMS 41 

<j>(m, z)=l + ~ g;-^:^2 + s -t P qp ... 

J. -tf O 1? 

/33\ ^ . ?M? 2 _ ra 3 a i . 

\ / o 9* "^ 

771 8 - ^Z. P 

~r" / * "^ i ~~ t " 

O i? i 

if we sum it by columns This double series is absolutely convergent 
[For, let \z\ = /> and \m\ = o- , if we replace each term by its absolute 
value, the sum of the terms of the new series included in the 
(p + l)th column is equal to 

o-Qr + 1) ( 



wliicli is the general teim of a convergent series We can therefore 
sum the double senes by rows, and we thus obtain for <t>(m, z) a 
development in power series 



From the relation (30) and the results established above ( 11), 
this series must be identical with that for e aim . Now for the coeffi- 
cient of m we have 



hence 1 

(34) 

where the determination of the logarithm to be understood is that 
one which becomes zero when z = 0. We can again represent the 
last expression by (1 + z) m , but in order to know without ambiguity 
the value in question, it is convenient to make use of the expression 



Let m = p + v i , if r and a have the same meanings as in the 
preceding paragraph, we have 

gmLog (1+ a) ___ ^(l* + it) (log r + tar) 

= e ftl sr ~ ^[cos (pa + v logr) + i sin (pa + v logr)]. 

In conclusion, let us study the series on the circle of convergence. Let U n 
"be the absolute value of the general term for a point z on the circle The ratio of 
two consecutive terms of the senes of absolute values is equal to | (m w+l)/n|, 
that is, if m = n + w, to 



42 ELEMENTS OF THE THEORY [I, 18 

where the function $(n) remains finite when n increases indefinitely By a 
known rule for convergence (I, 163) tins senes is conveigent when /i + 1 > 1 
and divergent in every other case The senes (29) is therefore absolutely con- 
vergent at all the points on the circle of convergence when AC is positive 

If fj, + 1 is negative or zero, the absolute value of the geneial teim never 
decreases, since the ratio U n +i/U n is nevei less than unity The senes w> diver- 
gent at all the points on the circle when /x =i 1. 

It remains to study the case where 1< /* =i Let us con&idei the series 
whose general term is U* , the ratio of two consecutive terms is equal to 



.*(")"!* i 
_-_ i i 

n 2 J 



and if we choose p laige enough so that p (/* + 1) > 1, this series will be conver- 
gent. It follows that J7, and consequently the absolute value of the geneial 
term l/ n , appi caches zero. This being the case, in the identity 



let us retain on each side only the terms of degree less than or equal to n , 
there remains the i elation 



^ 

where S n and S^ indicate respectively tho sum of the fix at (n + 1) toims of 
0(m, z) and of 0(m + 1, 2) If the ical pait of m lies between 1 and 0, the 
real part of m + 1 is positive Suppose | z \ = 1 , when the nuinbex n increases 
indefinitely, <8 approaches a limit, and the labt teim on the right appioachos 
zero , it follows that S n also approaches a limit, unless 1 + 2 = Theiefore, 
when 1< AC 35 0, the series is convergent at all the points on the circle of conver- 
gence, except at the point 2 = 1 



IIT CONFORMAL RE PRESENTATION 

19 Geometric interpretatxon of the derivative. Lot n = -V + Yl 1> a 

function of the complex variable , analytic within a closed curve C. 
We shall represent the value of n by the point whose coordinates aie 
JT, Y with respect to a system of rectangular axes. To simplify the 
following statements we shall suppose that the axes OX, Y are par- 
allel respectively to the axes Ox and Oy and ai ranged in the same order 
of rotation in the same plane or in a plane parallel to the plane trOy. 
When the point % describes the legion A bounded by tho closed 
curve C, the point u with the coordinates (-Y, F) describes in its 
plane a region A' , the relation u ==/() defines then a certain corre- 
spondence between the points of the two planes or of two portions of 
a plane On account of the relations which connect the derivatives of 
the functions X, F, it is clear that this correspondence should possess 
special properties. We shall now show that the angles are unchanged. 



I, 



COJSIFORMAL REPRESENTATION 



43 



Let # and ^ be two neighboring points of the region A, and u and 
Wj the coi responding points of the region A'. By the original defini- 
tion of the derivative the quotient (2^ u)/(^ z) has for its limit 
f(z) when the absolute value of x & approaches zeio in any 
manner whatever Suppose that the point l approaches the point 
2 along a cuive C, whose tangent at the point & makes an angle a 
with the paiallel to the direction Ox , the point ^ will itself de- 
scribe a cuive C' passing through u Let us discard the case in 
which f'(z) is zeio, and let p and <D be the absolute value and the 
angle of f(z) respectively Likewise let r and r' be the distances 
z^ and uu v a' the angle which the direction ^ makes with the 
parallel &x? to Ox, and ft' the angle which the direction uu^ makes 
with the paiallel uX' to OX The absolute value of the quotient 



\0' 





I) a 



FIG 96 



(7 a u)/( t z) is equal to i\/r, and the angle of the quotient is 
equal to ft' a'. We have then the two relations 



(35) 



liin (p a 1 } = 



2 kir 



Let us consider only the second of these relations. We may sup- 
pose k = 0, since a change in k simply causes an increase in the 
angle w by a multiple of 2?r. When the point ^ approaches the 
point & along the curve C, a 1 approaches the limit a, ft' approaches a 
limit ft, and we have ft = a + That is to say, in 01 der to obtain tJie 
direction of the tangent to the curve described by the point ir, it suffices 
to turn the direction of the tangent to the curve described by & through 
a constant angle o> It is naturally understood in this statement that 
those dnections of the two tangents are made to coriespond which 
correspond to the same sense of motion of the points & and u. 

Let D be another curve of the plane xOy passing through the point 
Zj and let D f be the corresponding curve of the plane XOY If the 
letters y and 8 denote zespectively the angles which the corresponding 



44 ELEMENTS OF THE THEORY [I, 19 

directions of the tangents to these two curves make with zx' and 
uX' (Figs, 9 a and 9 #), we have 

/3 = #-f<f>, S = y + o>, 

and consequently 8 /3 = y a The curves C' and D* cut each 
other in the same angle as the curves C and D Moreovei, we see that 
the sense of rotation is pieserved. It should be noticed that if 
/'(#) = 0, the demonstration no longei applies 

If, in particular, we considei, in one of the two planes xOy or XOY, 
two families of orthogonal curves, the corresponding curves in the 
other plane also will form two families of orthogonal cuives For 
example, the two families of curves X = C, Y = C', and the two 
families of curves 

(36) !/(*)!= C, angle /(*)=C" 

form orthogonal nets in the plane x 0y, for the corresponding curves 
in the plane XOY are, in the first case, two systems of paiallels to the 
axes of coordinates, and, in the other, circles having the origin for 
center and straight lines proceeding from the origin 

Example 1 Let z f = a?*, wheie a is a real positive numbei Indicating by 
r and 6 the polar coordinates of 2, and by r' and Q' the polai cooidmates of 2', 
the preceding relation becomes equivalent to the two lelations r' = ? a , 6' = a& 
"We pass then from the point z to the point 2' by raising the ladius vectoi to 
the powei a and by multiplying the angle by a The angles are pieseived, ex- 
cept those which have their vciticos at the ongin, and these aie multiplied by 
the constant factoi a 

Example 2 Let us consider the general linear transformation 



- 



= 



where a, 5, c, d are any constants whatever In ceitam paiticular cases it is 
easily seen how to pass from the point z to the point z'. Take for example the 
transformation zf = z + b , let 2 = to + yi"> %' == *' -f y't, b = a + /& , the picced- 
ing relation gives x' = x -f a, y' = 2/ + ft which shows that wo pass fiom the 
point z to the point z f by a translation 

Let now zf = az , if p and w indicate the absolute value and angle of a lospec- 
tively, then we have r f = /or, 6' = w + ^ Hence we pass f lorn the point 2 to the 
pomt^ by multiplying the radius vector by the constant factoi p and then turning 
this new radius vector through a constant angle w We obtain then the transfor- 
mation defined "by the formula z' = az by combining an expansion with a rotation 

Finally, let us consider the relation 



where r, 0, r x , 0' have the same meanings as above We must have rr' = 1, 
B + Q' = The product of the radu vectores is therefore equal to unity, while 



I, 20] CONFORMAL REPRESENTATION 45 

the polar angles aie equal and of opposite signs Given a circle O with center 
A and radius E, we shall use the expression mv&sion with respect to the given 
circle to denote the transformation by which the polar angle is unchanged but 
the radius vector of the new point is R 2 /r We obtain then the transformation 
defined by the relation z'z = 1 by carrying out first an inversion with respect to 
a circle of unit radius and with the origin as centei, and then taking the sym- 
metric point to the point obtained with respect to the axis Ox 

The most general transfoimation of the foim (37) can be obtained by com- 
bining the transformations which we have just studied If c = 0, we can replace 
the transformation (37) by the succession of transformations 

a , . b 

*, = -*, z' = z 1 + - 

If c is not zero, we can carry out the indicated division and write 

, __ a be ad 
Z ~" 



and the transformation can be leplaced by the succession of tiansformations 

*! = * + -* 2 2 = c 2 *i, 3 = - 
c z z 

z 4 = (6c ad) z s , z' = z 4 -f - 

c 

All these special transformations leave the angles and the sense of rotation 
unchanged, and change circles into circles Hence the same thing is then tiue 
of the general transformation (37), which is therefore often called a circular 
transformation In the above statement straight lines should be regarded as 
circles with infinite radii 
Example 3 Let 

Z 7 = (z - etf*i (z - e^ (z - %,)*, 

where e 1? e 2 , , % are any quantities whatever, and where the exponents %, 
m 2 , , nip are any real numbers, positive or negative Let If, J? t , jS? 2 -^P 
be the points which represent the quantities 2, e v e^ - , p , let also r 3 , r 2 , - 7 
r p denote the distances JfJBfj, 3O? 2 , , ME P and a , 2 , , P the angles which 
IS^ Jf, jE? 2 Jf, , E P M make with the parallels to Ox The absolute value and 
the angle of z' are respectively r^rj** - * r/>v and m^ + m 2 2 + 
Then the two families of curves 



form an orthogonal system When the exponents m t , m a , , ra p are rational 
numbers, all the curves are algebraic If, for example, p = 2, m t = m 2 = 1, one 
of the families is composed of Cassmian ovals with two foci, and the second 
family is a system of equilateral hyperbolas 

20. Confonnal transformations in general. The examination of the 
converse of the proposition which we have just established leads us to 
treat a more general problem Two surfaces, 2, S', being given, let 
us set up between them any point-to-point correspondence whatever 



46 



ELEMENTS OF THE THEORY 



LI, 20 



(except for certain broad restrictions winch will be made latei), 
and let us examine tlie cases in which the angles are unaltered in 
that transformation Let x, y, * be the rectangular coordinates of 
a point of S, and let x\ y f , ' be the lectangular cooidmates of a 
point of 3' We shall suppose the six coordinates x, y, z, x 1 , y\ r 
expiessed as functions of two variable parameters it, v in such a way 
that corresponding points of the two surfaces coi respond to the same 
pair of values of the parameters u, v 



(38) 




s'-U' = . 



'(*, ), 



Moreover, we shall suppose that the functions /, <, , together with 
their paitial derivatives of the first order, aie continuous when the 
points (SB, y, s) and (a;', y\ z 1 ) remain in ceitain legions of the two 
surfaces 2 and 5' We shall employ the usual notations (I, 131) 



(39) 



-<&' '-' *-><$$ 



2 Fdu dv 
JB'du* + ZF'du di) + G'dv*. 



Let C and D (Figs 10 a and 10 J) be two cuives on the stufacc ^, 
passing through a point m of that suiface, and C' and />' the corre- 
sponding curves on the surface 5' passing through the point w,' 




FIG 10 a 




. 106 



Along the curve C the parameters u t v are functions of a single 
auxiliary variable t, and we shall indicate their differentials by du, 
and dv Likewise, along D, u and v are functions of a variable ', and 
we shall denote their diiferentials here by 8u and Sw In general, we 
shall distinguish by the letters d and 8 the differentials relative to 
a displacement on the curve C and to one on the curve D. The 



I, 20] CONFORMAL REPRESENTATION 47 

following total diffeientials are pioportional to the direction cosines 
of the tangent to the curve C, 

, dx , dx _ _ Bt/ T a?/ _ 7 da . dx J 
dx = du + -%-dv, dy = ^-du + -^dv, dz du + -z-dv, 
on cv y cu ov cu ov 

and the following are proportional to the direction cosines of the 
tangent to the curve D, 



, 

ov ' du tic 

Let <o be the angle between the tangents to the two curves C and 
D The value of cos o> is given by the expiession 

dx Bx + d y 8y 4- dz $z 

cos w - J J -- 



^dx 2 + dif + 

which can be written, making use of the notation (39), in the form 
Edu &/ -f F(du Sv + du 8?Q + Gdv $v 



(40) cos o> = 



+ 2Fdu dv + Gdv z VE Sit? -f 2 FStc v -f G Sv 3 



If we let o) f denote the angle between the tangents to the two 
curves C' and Z> f , we have also 

,. , E'du 8z/ + F'(du S?; -f dv Bit} + G'clv Bv 

(41) COS a/ = / - 

^ y 



In order that the transformation considered shall not change the 
value of the angles, it is necessary that cos o>' = cos o>, whatever du, 
dv, Sit, Sv may be The two sides of the equality 

cos 2 a/ = cos 2 CD 

are rational functions of the latios Sv/Sit, dv/du, and these functions 
must be equal whatever the values of these ratios Hence the corre- 
sponding coefficients of the twp fractions must be proportional , that 
is, we must have 



where X is any function whatever of the parameters it, v These 
conditions are evidently also sufficient, for cos o>, for example, is a 
homogeneous function of E, F, G, of degree zero 

The conditions (42) can be replaced by a single relation ds = X% 2 , 01 

(43) ds' = \ds. 



48 ELEMENTS OF THE THEORY [I, 20 

This relation states that the ratio of two coi responding infinitesimal 
arcs approach a limit independent of du and of dv 9 when these two 
arcs approach zero This condition makes the reasoning almost 
intuitive Foi, let abc be an infinitesimal triangle on the first surface, 
and a'W the coi responding triangle on the second surface Imagine 
these two curvilineai triangles replaced by rectilinear triangles that 
approximate them Since the ratios a'V/ab, a'c'/ac, !>'c'/t>c approach 
the same limit X(M, v), these two triangles approach similarity and 
the corresponding angles appioaeh equality 

We see that any two corresponding infinitesimal figures on the 
two surfaces can be consideied as similar, since the lengths of the 
aics are pioportional and the angles equal , it is on this account that 
the term conformal representation is often given to every correspond- 
ence which does not alter the angles 

Given two surfaces 2, S' and a definite relation which establishes 
a point-to-point conespondence between these two surfaces, we can 
always determine whether the conditions (42) are satisfied or not, 
and therefore whether we have a conformal lepresentation of one 
of the surfaces on the other 

But we may consider other problems For example, given the sur- 
faces S and S', we may propose the problem of determining all the 
coirespondences between the points of the two surfaces which pre- 
seive the angles Suppose that the coordinates (x, y, z) of a point 
of 5 are expressed as functions of two parameters (, v), and that 
the coordinates (x', y', *) of a point of S' are expressed as functions 
of two other parameters (u' 9 v') Let 

?, ds a = E f du'* + 2 F f du 1 dv' 



be the expressions for the squares of the lineai elements The prob- 
lem in question amounts to this To find two functions u' = TT^M, -y), 
v 1 = 7T 2 (w, v) suck that we have identically 

d'7r i + G l d-rr\ = \\E du* + 2 F du, dv 



X being any function of the variables u, v The geneial theory of dif- 
ferential equations shows that this problem always admits an infinite 
number of solutions , we shall consider only certain special cases 

21. Conformal representation of one plane on another plane. Every 
correspondence between the points of two planes is defined by 
relations such as 

(44) X = P(x,y), F=Q(^y), 



I, 21] COISTFORMAL REPRESENTATION 49 

where the two planes are referred to systems of rectangular coordi- 
nates (x , ?/) and (A", F) From what we have just seen, in order that this 
tiansformation shall preseive the angles, it is necessaiy and sufficient 

that we have 

<LY* + dY* = \\dx* + df), 



where X is any function whatever of x 9 y independent of the differ- 
entials Developing the differentials dX, <#Fand comparing the two 
sides, we find that the two functions P(x, y) and Q(r, y) must 
satisfy the two relations 



dx dy dx dy 

The partial derivatives dP/dy, dQ/dy cannot both be zero, for the 
first of the relations (45) would give also dQ/dx = dP/dx = 0, and 
the functions P and Q would be constants. Consequently we can 
write according to the last relation, 

<9P_- 3Q 2Q__ &P 
dx dy dx dy 

where /-i is an auxiliary unknown. Putting these values in the first 
condition (45), it becomes 



and from it we derive the result ;LC = 1. We must then have 
either 

dP dQ, dP 3Q 

(4b) 5~" == V -.= 

^ ' ox uy oy ox 

or 

(47) ^^^^Q, ^ = 5. 

^ ' dx dy dy dx 

The first set of conditions state that P + Qi is an analytic func- 
tion of x + yi As for the second set, we can reduce it to the first 
by changing Q to Q, that is, by taking the figure symmetric to the 
transformed figure with respect to the axis OX Thus we see, finally, 
that to every conformal representation of a plane on a plane there 
corresponds a solution of the system (46), and consequently an 
analytic function If we suppose the axes OX and OY parallel re- 
spectively to the axes Ox and Oy, the sense of rotation of the angles 
is preserved or not, according as the functions P and Q satisfy the 
relations (46) or (47) 



50 



ELEMENTS OF THE THEORY 



[I, 22 



22. Riemann's theorem Given in the plane of the vanable * a region A 
bounded by a single curve (or simple boundary), and in the plane of the vari- 
able u a circle C, Riemann proved that there exists an analytic function u = /(*), 
analytic in the legion A, such that to each point of the legion A conesponds 
a point of the cncle, and that, concisely, to a point of the oiiole conesponds 
one and only one point of A The function /() depends also upon thiee 
aibitiaiy leal constants, which we can dispose of in such a way that the center 
ot the cncle coiiesponds, to a given point of the region A, while an aibitianly 
chosen point on the cncuinfeience corresponds to a given point of the boundaiy 
of A We shall not give here the demonstiation of this theorem, of which we 
shall indicate only some examples 

We shall point out only that the circle can be replaced by a half -plane 
Thus, let us suppose that, in the plane of u, the circumference passes through the 
origin , the transfoimation u' = 1/u lepJaces that circumference by a straight 
line, and the circle itself by the poition of the w'-plane situated on one side ot 
the stiaight line extended indefinitely in both directions 

Example 1 Let u = z 1 /*, where a is real and positive Consider the portion 
A of the plane included between the direction Ox and a ray through the origin 
making an angle of cm with Ox (a ^ 2). Let z = re*, u = R& , we have 

B-4 w =- 

~~ ' "" at 

When the point z describes the portion A of the plane, r varies from to 
f oo and 9 from to air , hence R varies fiom to + oo and w fiom to it 



y ' i 



! 



it 




FIG. 11 



The point u therefore describes the half -plane situated above the axis OX, and 
to a point of that half -plane corresponds only one point of -4, for we have, 
inversely, r = It", Q = a<a 

Let us next take the portion B of the #-plane bounded by two arcs of circles 
which intersect Let 3 , z l be the points of intersection , if we carry out first the 
transformation 



the region B goes over into a portion A of the emplane included between two 
rays from the origin, for along the arc of a -circle passing through the points 



I, 22] CONFORMAL REPRESENTATION 51 

2 , z lt the angle of (z z )/(z 2 X ) lemams constant Applying now the pre- 
ceding tianstormation u = (z') l /, we see that the function 



enables us to lealize the conformal lepiesentation of the region B on a half- 
plane by suitably choosing a 

Example 2 Let u = cos 2 Let us cause z to describe the infinite half -strip 
JB, or AOBA' (Fig 11), defined by the inequalities ^ x ^ IT, y a 0, and let 
us examine the legion described by the point u = X + Yi We have here ( 12) 



(48) 



When a; varies fiom to ir, F is always negative and the point u remains in 
the half -plane below the axis X'OX Hence, to eveiy point of the region E 
corresponds a point of the u half-plane, and when the point z is on the bound- 
aiy of JR, we have T = 0, for one of the two factois sin x, 01 (& e-v)/2 is zeio 
Conversely, to every point of the u half -plane below OX corresponds one and 
only one point of the strip E in the s-plane In fact, if z' is a root of the equa- 
tion u cos 2, all the other roots are included in the expression 2 Tcir z* If 
the coefficient of ^ in z f is positive, there cannot be but one of these points in the 
strip E, for all the points 2 kv z' are below Ox There is always one of 
the points 2 for + zf situated in JR, for there is always one of these points whose 
abscissa lies between and 2 TT That abscissa cannot be included between TT 
and 2 TT, for the corresponding value of Y would then be positive The point is 
therefore located in E 

It is easily seen fiom the formulae (48) that when the point z describes the 
portion of a paiallel to Ox in JR, the point u describes half of an ellipse When 
the point z describes a parallel to Oy, the point u describes a half-branch of a 
hyperbola All these conies have as foci the points C, G" of the axis O.X, with 
the abscissas + 1 and I 

Example 3 Let vz 



(49) 



where a is real and positive In order that \u\ shall be less than unity, it is 
easy to show that it is necessary and sufficient, that cos [(7r^)/(2 a)] > If y 
vanes from a to + a, we see that to the infinite stup included between the 
two straight lines y = a, y = + a corresponds in the w-plane the circle C 
described about the origin as center with unit radius Conversely, to every 
point of this circle corresponds one and only one point of the infinite stnp, for 
the values of z which correspond to a given value of u form an arithmetical pro- 
gression with the constant difference of 4 ai Hence there cannot be more than 
one value of z in the stnp considered Moreover, there is always one of these 
roots in which the coefficient of i lies between a and 3 a, and that coefficient 
cannot he between a and 3 a, for the corresponding value of | u | would then be 
greater than umty 



52 ELEMENTS OF THE THEOEY [I, 23 

23. Geographic maps. To make a confoimal map of a surface 
means to make the points of the surface coi respond to those of a 
plane in such a way that the angles are unalteied Suppose that the 
cooidmates of a point of the surface S undei consideration be ex- 
pressed as functions of two variable parameters (u, v), and let 

-f- 2 Fdu dv + G dv 2 



be the square of the linear element for this surface Let (a, ft) be 
the lectangular coordinates of the point of the plane P which cor- 
responds to the point (u, v) of the surface The problem here is to 
find two functions 

w = 7^(0,0), * = ir fl (*,) 

of such a nature that we have identically 



where A. is any function whatever of <x y fi not containing the differ- 
entials This problem admits an infinite number of solutions, which 
can all be deduced from one of them by means of the conformal 
tiansformations, already studied, of one plane on another Suppose 
that we actually have at the same time 

df = \(d<P + dfP), ds* = X f (da* + dp*) ; 
then we shall also have 

da* + df? = ~ (<fo a + dp*), 

A 

so that a + pi, or a pi, will be an analytic function of a* + p'i 
The converse is evident 

Example 1 Mercator's projection We can always make a map of a 
surface of revolution in such a way that the meridians and the paral- 
lels of latitude correspond to the parallels to the axes of coordinates 
Thus, let 



be the coordinates of a point of a surface of revolution about the 
axis 0& , we have 



which can be written 
if we set 




I, 23] CONFORMAL REPRESENTATION 53 

In the case of a sphere of radius R we can write the coordinates in 
the form 

x = R sin cos <, y = R sin 6 sin <, & = J2 cos 0, 



and we shall set 



We obtaui thus what is called Mercator's projection, in which the 
meridians are represented by parallels to the axis OY, and the paral- 
lels of latitude by segments of straight lines parallel to OX. To 
obtain the whole surface of the sphere it is sufficient to let <f> vary 
from to 2 TT, and from to TT , then A" varies from to 2 TT and Y 
from oo to + oo The map has then the appearance of an infinite 
strip of breadth 2 TT The curves on the surface of the sphere which 
cut the meridians at a constant angle are called loxodromio curves 
or rhumb lines, and are represented on the map by straight lines. 
Example 2. Stereo graphic projection. Again, we may write the 
square of the linear element of the sphere in the form 




or 

ds* = 4 cos 4 1 (dp* + p^o 2 ), 
if we set 

Q 

p = jR tan jr ? co =$. 



But efy> a + /) 2 ^w 2 represents the square of the linear element of the 
plane in polar coordinates (p, o>) ; hence it is sufficient, in order to 
obtain a conf ormal representation of the sphere, to make a point of 
the plane with polar coordinates (p, o>) correspond to the point (0, <) 
of the surface of the sphere It is seen immediately, on drawing the 
figure, that p and <o are the polar coordinates of the stereograpluc 
projection of the point (0, <) of the sphere on the plane of the 
equator, the center of projection being one of the poles * 

* The center of prelection is the south pole if 6 is measured from the north pole 
to the radius Using the north pole as the center of projection, the point (IP/p, w), 
symmetric to the first point (see Ex 17, p 58), would be obtained TRANS 



54 ELEMENTS OF THE THEORY [I, 23 

Example 3 Map of an anchor ring Consider the anchoi ring generated by 
the revolution of a circle of ladius R about an axis situated in its own plane at 
a distance a from its centei, wheie a > R Taking the axis of i evolution for the 
axis of 3, and the median plane of the anchor ring foi the icy-plane, we can 
wnte the coordinates of a point of the surface in the foim 

x = (a 4- E cos 0) cos 0, y = (a + It cos0)sm0, z = #sm0, 
and it is sufficient to let 8 and $ vaiy from ir to + if From these formulae 
we deduce 




and, to obtain a map of the suiface, we may set 



where 



Thus the total surface of the anchor ring corresponds point by point to that 
of a rectangle whose sides are 2 TT and 2 we/ Vl e 2 

34. Isothermal curves Let U(x, y) be a solution of Laplace's equation 



the curves represented by the equation 

(60) U(x, y) = C, 

where C7 is an aibitraiy constant, form a family of isothermal curves With every 
solution &(x, y) of Laplace's equation we can associate another solution, 
"7(05, y), such that U + Viis an analytic function of x + yi The relations 

8CT = 3F ^ = _.Z 
d& dy dy dx 

show that the two families of isothermal curves 

0>, y) = 0, F(fc, y) = C' 

are orthogonal, for the slopes of the tangents to the two curves C and C' are 
respeomely _S_8U > _8F_F 

3aj 5y 2x dy 

Thus the orthogonal trajectories of a family of isothermal curves form another 
family of isothermal curves We obtain all the conjugate systems of isothermal 
curves by considering all analytic functions f(z) and taking the curves for 
which the real part of f(z) and the coefficient of % have constant values The 
curves for which the absolute value E and the angle O of f(z) remain constant 
also form two conjugate isothermal systems , for the real part of the analytic 
function Log [/()] is log 22, and the coefficient of i is Q 

Likewise we obtain conjugate isothermal systems by considering the curves 
described by the point whose coordinates are JT, F, where f(z) = X + Yi, when 



I, 24] COHFORMAL REPRESENTATIONS 55 

we give to x and y constant values. This is seen by regarding x -f- yi as an 
analytic function of J5T+ Yi More generally, every tiansfoiination of the 
points of one plane on the other, which preserves the angles, changes one family 
of isothermal curves into a new family of isothermal curves Let 



be equations defining a transformation which preserves angles, and letF(o;', y') 
be the result obtained on substituting p (x', y') and q (x', tf) for x and y in 27(0;, y) 
The proof consists in showing that F(JC', 2/0 is a solution of Laplace's equation, 
provided that U(x, y) is a solution The verification of this fact does not offer 
any difficulty (see Vol I, Chap III, Ex 8, 2d ed , Chap II, Ex 9, 1st ed ), 
but the theorem can be established without any calculation Thus, we can sup- 
pose that the functions p (#', y^) and q (x', ]/) satisfy the relations 



for a symmetric transformation evidently changes a family of isothermal curves 
into a new family of isothermal curves. The function x + yi = p + qi is then 
an analytic function of zf = tf + y% and, after the substitution, IT 4- Vi also 
becomes an analytic function F(x', y*) + v$ (&', y') of the same variable z' 
( 5) Hence the two families of curves 



=0, $(',20 = 0' 
give a new orthogonal net f oimed by two corrugate isothermal families. 

Ifoi example, concentric cucles and the rays from the center form two con- 
jugate isothermal families, as we see at once by considering the analytic func- 
tion Log z Carrying out an inversion, we have the result that the circles 
passing through two fixed points also form an isothermal system. The conjugate 
system is also composed of circles 

Likewise, conf ocal ellipses form an isothermal system. Indeed, we have seen 
above that the point u = cos z descnbes conf ocal ellipses when the point z is 
made to descnbe parallels to the &xis Ox ( 22) The conjugate system is made 
up of conf ocal and orthogonal hyperbolas 

Note In order that a family of curves represented by an equation P (, y) = C 
may be isothermal, it is not necessary that the function P (, y) be a solution of 
Laplace' s equation Indeed, these curves are represented also by the equation 
0[P(ai, y)] = 0, whatever be the function <p , hence it is sufficient to take for 
the function a form such that U(x, y) = 4>(P) satisfies Laplace's equation 
Making the calculation, we find that we must have 



hence it is necessary that the quotient 

3 2 P gP 



depend only on P, and if that condition is satisfied, the function can be 
obtained by two quadratures. 



56 ELEMENTS OF THE THEORY [I, Exs 

EXERCISES 

1 Determine the analytic function f(z) = X + Ti whose real part X is 
equal to 2sm2s 



Consider the same question, given that -"+ T is equal to the preceding 
function 

2 Let (m, p) = be the tangential equation of a real algebraic curve, that 
is to say, the condition that the straight line y = ma; -f P be tangent to that 
curve The roots of the equation (&, zi) = are the real foci of the curve 

3 If p and q are two integers prime to each other, the two expressions 
(Vz) p and VZP are equivalent What happens when p and q have a greatest 
common divisor d > I ? 

4 Pmd the absolute value and the angle of e^ + J^ by considering it as 
the limit of the polynomial [1 + (x + yi)/m\ m when the integer m increases 
indefinitely 

5. Piove the formulae /n + 1 A 

cos a + cos(a + 6) -f + cos(a + rib) = cos (a + ) , 

sm(-^ X 2/ 



I n5 

sina + am (a + 6) + - + sm (a + rib) = - sin la + y 



6. What is the final value of arc sin z when the variable z describes the seg- 
ment of a straight line from the origin to the point 1 + *, if the initial value of 
arc sm z is taken as 9 

7. Prove the continuity of a power series by means of the formula (12) ( 8) 

/( + *)-/() = A/! (2) + J5/,<) + + 5/( Z > + 

[Take a suitable dominant function for the series of the right-hand side ] 

8 Calculate the integrals 

i x m QOX C0 g ftp ^ Cym 000; SIn fo> $, 

fctn(cc a) ctn(x~ 6) - ctn(sc Z)dcc 

9 Given in the plane xOy a closed curve (7 having any number whatever of 
double points and described in a determined sense, a numerical coefficient is 
assigned to each region of the plane determined by the curve according to the rule 
of Volume I ( 97, 3d ed , 96, 1st ed) Thus, let .K, B' be two contiguous regions 
separated by the arc ab of the curve described in the sense of a to 6 , the coeffi- 
cient of the region to the left is greater by unity than the coefficient of the 
region to the right, and the region exterior to the curve has the coefficient 



I,Exs] EXERCISES 57 

Let ZQ be a point taken in one of the regions and N the corresponding coeffi- 
cient Prove that 2Nv represents the variation of the angle of z z ^hen 
the point z describes the curve C m the sense chosen 

10 By studying the development of Log[(l+ z)/(l 2)] on the circle of 
convergence, prove that the sum of the series 

smfl sm80 sm50 sin (2 u 



1 3 5 2n+l 

is equal to ?r/4, according as sm 6 > (Cf Vol I, 204, 2d ed , 198, 1st ed ) 

11 Study the curves described by the point Z = z z when the point z describes 
a straight line or a circle 

12 The relation 2Z = z + c*/z effects the conformal representation of the 
region inclosed between two conf ocal ellipses on the ring-shaped region bounded 
by two concentric circles 

[Take, for example, z = Z -f Vz 2 c 2 , make m the Z-plane a straight-line 
cut ( c, c), and choose for the radical a positive value when Z is real and 
greater than c ] 

13. Every circular transformation z' = (az + b)/(cz + d) can be obtained by 
the combination of an eoen number of inversions Prove also the converse 

14 Eveiy transformation defined by the relation zf = (az + b)/(cz + d), 
where Z Q indicates the conjugate of z, results from an odd number of inversions 
Prove also the converse 

15. Fuchsian transformations. Every linear transformation ( 19, Ex 2) 
z' = (az + b)/(cz + d), where a, 6, c, d are real numbers satisfying the relation 
ad 6c = 1, is called a Fuchsian transformation Such a transformation sets 
up a correspondence such that to every point z situated above Ox corresponds a 
point z' situated on the same side of Ox' 

The two definite integrals 



/ 



dxdy 

-- 



are invariants with respect to all these transformations 

The preceding transformation has two double points which correspond to 
the roots or, of the equation as 2 + (d a)2 6 = If a and are real and 
distinct, we can write the equation z' = (az + V)/(cz + d) m the equivalent form 



where k is real Such a transformation is called hyperbolic. 

If cc. and $ are conjugate imaginan.es, we can write the equation 



where is real Such a transformation is called elliptic 
If j8 = a, we can write 



where a and k are real Such a transformation is called parabola. 



58 ELEMENTS OF THE THEORY [I,Exs 

16 Let z' =/() be a Fuchsian transformation Put 



Prove that all the points z,z 1 ,z 2 , , z n are on the circumf eience of a circle. 
Does the point z n approach a limiting position as n increases indefinitely ? 

17. Given a circle C with the center O and radius JR, two points 3f, JfcP 
situated on a ray fiom the center are said to be symmetric with respect to 
that circle if OM x OM' = R* 

Let now C, C' be two cncles in the same plane and M any point whatevei 
in that plane Take the point Jlfj symmetric to M with respect to the circle (7, 
then the point M { symmetric to M with respect to C", then the point Jf 2 sym- 
metric to M{ with respect to C, and so on forever Study the distribution of the 
points .M^, -M"i, Jf 2 , Jj, " 

18. Find the analytic function Z=f(z) which enables us to pass from 
Mercator's projection to the stereogiaphic projection 

19* All the isothermal families composed of circles are made up of circles 
passing through two fixed points, distinct 01 coincident, real or imaginary 

[Setting z = x 4- 2/1, Z Q = x yi, the equation of a family of circles depending 
upon a single parameter X may be written in the form 

ZZ Q + az + bz + c = 0, 

where a, 6, c are functions of the parameter X In order that this family be 
isothermal, it is necessary that d z \/dzdz = Making the calculation, the 
theorem stated is proved ] 

20*. If |g| < 1, we have the identity 



[EULER ] 

[In order to prove this, transform the infinite product on the left into an infinite 
product with two indices by putting in the first row the factors 1 + g, 1 + g 2 , 
1 + g 4 , , l + g 2 ", , m the second row the factors 1 + g s , 1+9 6 , i 
1 + (g s ) 2n , - ; and then apply the formula (16) of the text ] 

21. Develop in powers of z the infinite products 

F(z) = (1 + xz) (1 + x*z) (1 + xz) , 
*(z) = (1 -f xz) (1 -f a^) (l + x *+iz) .... 

[It is possible, for example, to make use of the relation 
F(xz) (1 + xz) = F(z), *(x*z) (I + xz) = 
22*. Supposing \x\ < 1, prove Euler's formula 



(See J. BBBTKAND, (7afcuZ d^rewiicZ, p 328 ) 



I, Exs ] EXERCISES 59 

23* Given a sphere of unit radius, the stereographic projection of that sphere 
is made on the plane of the equatoi, the center of projection being one of the 
poles To a point M of the sphere is made to correspond the complex number 
s = x + yi, where x and y are the lectangular coordinates of the projection m of 
M with lespect to two rectangular axes of the plane of the equator, the origin 
being the center of the sphere To two diametrically opposite points of the 
sphere coriespond two complex numbers, s, l/s , where s is the conjugate 
imaginary to 8 Every linear transformation of the form 



(A) 



where p<x + 1 = 0, defines a rotation of the sphere about a diameter. To groups 
of rotations which make a regular polyhedron coincide with itself correspond 
the groups of finite order of linear substitutions of the form (A). (See KLEIN, 
Das Ikosaeder ) 



CHAPTER II 

THE GENERAL THEORY OF ANALYTIC FUNCTIONS 
ACCORDING TO CAUCHY 

I DEFINITE INTEGRALS TAKEN BETWEEN 
IMAGINARY LIMITS 

25. Definitions and general principles. The results presented in the 
preceding chapter are independent of the work of Cauchy and, for 
the most part, prior to that woik We shall now make a system- 
atic study of analytic functions, and determine the logical conse- 
quences of the definition of such functions Let us recall that a 
function f(z) is analytic in a region A . 1) if to every point taken 
in the region A corresponds a definite value of f(z) ; 2) if that 
value varies continuously with & ; 3) if for every point taken in A. 
the quotient / + &)-/() 

h 

approaches a limit f(z) when the absolute value of h approaches zero. 

The consideration of definite integrals, when the variable passes 
through a succession of complex values, is due to Cauchy * ; it was 
the origin of new and fruitful methods. 

Let f() be a continuous function of 2 along the curve A MB 
(Fig. 12) Let us mark off on this curve a certain number of points 
of division , v 3 2 , , s n _ 1? 2', which follow each other in the order 
of increasing indices when the arc is traversed from A to B, the 
points & and #' coinciding with the extremities A and B 

Let us take next a second series of points 1? 2 , -, n on the arc 
AB, the point & being situated on the arc *|._i* t and let us consider 
the sum 



+/k) (** - **-0 + +/( <*' - *-0 

When the number of points of division 1? , n-1 increases indefi- 
nitely in such a way that the absolute values of all the differences 

* Memoire sur les integrates defimes, prises entre des hrmtes ^mag^na^res, 1825 
This memoir is reprinted m Volumes VII and VIII of the Bulletin des Sciences math& 
matiques (1st series) 

60 



n, 25] 



DEFINITE INTEGRALS 



61 



z i ~~ *o> ^2 ~~ *i> beeome and remain smaller than any positive 
number arbitrarily chosen, the sum S approaches a limit, which is 
called the definite integial of /() taken along AMB and which is 
represented by the symbol 



L 



(.AMB) 



To prove this, let us separate the real part and the coefficient of i 
in S, and let us set 




FIG 12 

where X and Y are continuous functions along AMB. Uniting the 
similar terms, we can write the sum S in the form 

l - fl >+" 



] 

When the number of divisions increases indefinitely, the sum of the 
terms in the same row has for its limit a line integral taken along 
AMJB, and the limit of S is equal to the sum of four line integrals:* 



f /()<& = f (Xdx - Ydy) 4- i T 

J(AMS) JuilB) JUMB) 



* In order to avoid useless complications in the proofs, we suppose that the coor- 
dinates x t y of a point of the arc AMB are continuous functions x= $ (<), y ~ $ (2) of 
a parameter t, which have only a finite number of maxima and minima "between A 
and B We can then hreak up the path of integration into a finite numoer of arcs 
which are each i epresented by an equation ol the form y*=F(ti), the function F being 
continuous between the corresponding limits , or into a finite number of arcs which 
are each represented by an equation of the form fc= G (y) There is no disadvantage 
in making this hypothesis, for in all the applications there is always a certain amount 
of freedom in the choice of the path of integration Moreover, it would suffice to 
suppose that (2) and V ($) are functions of limited vanation We have seen that 
in this case the curve AMB is then rectifiable (I, ftns , 73, 82, 95, 2d ed ). 



62 THE GENERAL CAUCHY THEORY [H,25 

From the definition it results immediately that 
f f(*)d*+C /(*)<fe = 

J(AMB) J(BMA) 

It is often important to know an upper bound for the absolute value 
of an integral Let s be the length of the arc AM, L the length of 
the arc AB, s L _ l9 s^, ^ the lengths of the arcs A^_ ly Az L , A& of 
the path of integration Setting F(s) = |/() |, we have 

|/(&)(*i - **-i) I = F (*d K - *i-i| = F (*d ( s i - **-0 
for | t ^_ 1 | represents the length of the chord, and S L S L _ I the 
length of the arc. Hence the absolute value of S is less than or at 
most equal to the sum ^F(a- k )(s L s*,_i) , whence, passing to the 
limit, we find , r r z 

I /(.)& S / F(s)ds. 

I J(AMB) */0 



Let M be an upper bound for the absolute value of /() along the 
curve AB. It is clear that the absolute value of the integral on the 
right is less than ML, and we have, a fortiori, 



jL 



f(*)d* 



<ML. 



26. Change of variables. Let us consider the case that occurs fre- 
quently in applications, in which the coordinates #, y of a point of 
the arc AB are continuous functions of a variable parameter t, 
x = <f> (), y = if/ (), possessing continuous derivatives <' (t), ij/' (t) ; and 
let us suppose that the point (x, y) describes the path of integra- 
tion from A to B as t varies from a to ft Let P() and Q(f) be the 
functions of t obtained by substituting <() and i^(tf), respectively, 
for x and y in X and F 

By the formula established for line integrals (I, 95, 2d ed.; 93, 
1st ed.) we have 



X/0 
Xdx Ydy = I 
15) Jet 

Xr^ 
JS"^ + Ydx =s I [P(#) i/r'() + Q(#) ^ f (#)]^. 
UB) Jet 



Adding these two relations, after having multiplied tlie two sides 
of the second by *, we obtain 



(1) I f()dz= C [P(tf) 
*/U,B) */* 



II, 26] DEFINITE INTEGRALS 63 

This is precisely the result obtained by applying to the integral 
ff(z) dz the formula established for definite integrals in the case of 
real functions of leal variables, that is, in order to calculate the 
integral ff(&)dz we need only substitute <j>(f) + i\//(t) for & and 
[>'(*) + fy f (*)]<** for d * m/(s)d* The evaluation of ff(*)d* is 
thus reduced to the evaluation of two ordinary definite integrals. If 
the path A MB is composed of several pieces of distinct curves, the 
formula should be applied to each of these pieces separately. 

Let us consider, for example, the definite integral 



r +1 dz 
J-i 



We cannot integrate along the axis of reals, since the function to be 
integrated becomes infinite for # = 0, but we can follow any path 
whatever which does not pass thiough the origin Let & describe a 
semicircle of unit radius about the origin as center. This path is 
given by setting % == e tl and letting t vary from TT to 0. Then the 
integral takes the form 



/* l dz r Q r Q r Q 

~= I ie-*dt=*il co8td+ I sm<& = 2. 

This is precisely the result that would be obtained by substituting 
the limits of integration directly in the primitive function 1/z 
according to the fundamental formula of the integral calculus 
(I, 78, 2d ed , 76, 1st ei). 

More generally, let z = <f> (u) be a continuous function of a new complex 
variable u = + i\i such that, when u describes in its plane a path C2VD, the 
variable describes the curve AMB To the points of division of the curve 
AMB correspond on the curve GND the points of division u , Uj, w 2 , - , uj._i, 
ui , ,u' If the function <j> (u) possesses a derivative <f>'(u) along the curve CND y 
we can write 

*L - *-! 



where e^ approaches zero when KI approaches U L ~I along the curve CND 
Taking ^t i = ^jt i an< i replacing z% ZLI by the expression derived from the 
preceding equality, the sum $, considered above, becomes 



8 = 

A 

The first part of the right-hand side has for its limit the definite integral 

J(C2a>) 



64 THE GENERAL CAUCHY THEORY [n, 26 

As for the remaining term, its absolute value is smaller than t\ML', where 17 is a 
positive number greater than each of the absolute values | e |and where Z' is the 
length of the curve CND If the points of division can be taken so close that 
all the absolute values ] ej, | will be less than an arbitrarily chosen positive num- 
ber, the remaining term will approach zero, and the general formula for the 
change of variable will be 

(2) C f(z) dz = f f[<t> (u)] <t>'(u) du 

^ ' J(AMB) J(CND) 

This formula is always applicable when (u) is an analytic function , in fact, 
it will be shown later that the derivative of an analytic function is also an 
analytic function* (see 34) 

27. The formulae of Weierstrass and Darboux. The proof of the law 
of the mean f 01 integrals (I, 76, 2d ed , 74, 1st ed ) rests upon 
certain inequalities which cease to have a precise meaning when 
applied to complex quantities Weierstrass and Darboux, however, 
have obtained some interesting results in this connection by con- 
sidering integrals taken along a segment of the axis of reals We 
have seen above that the case of any path whatever can be reduced 
to this particular ease, provided certain mild restrictions are placed 
upon the path of integration. 

Let / be a definite integral of the following form . 



r =* f 

Ja 



* If this property is admitted, the following proposition can easily be proved 
Letf(z) be an analytic function in a finite region A of the plane For every pos^ 
tive number e another positive number ij can be found such that 

r (*)!<*, 



when z and z + h are two points of A whose distance from each other \h\is less than t\ 
For, let/ (2) = P (a;, y) + iQ (aj, y) , h = A + 1 Ay From the calculation made m 3, to 
find the conditions for the existence of a unique derivative, we can write 

, , _ t 

J(Z) ~ 



^ [P' y (x + As, y + 0Ay) - P' y (a, y)] Ay 

ASB + i&y 

+ 

Since the derivatives P^, Py , Q& Qy are continuous m the region A, we can find a num- 
ber 17 such that the absolute values of the coefficients of A* and of Ay are less than e/4, 
when VAi; 2 + Ay 2 is less than ij Hence the inequality written down above will be 
satisfied if we have ( h | < ij This being the case, if the function <f> (u) is analytic in 
the region A, all the absolute values | e* | will be smaller than a given positive number e, 
provided the distance between two consecutive points of division of the curve 
is less than the corresponding number 17, and the formula (2) will be established 



II, 27] DEFINITE INTEGRALS 65 



where /(), <j>(t), $(t) are three real functions of the real variable t 
continuous in the interval (a, ft). From the veiy definition of the 
integral we evidently have 

1 = f /(*) < 09 dt + * f 

*/<r ,Ar 

Let us suppose, for defimteness, that a < ft , then a is the length 
of the path of integration measured from a, and the general formula 
which gives an upper bound for the absolute value of a definite 
integral becomes 



or, supposing that/() is positive between a and 
\i\s 



Applying the law of the mean to this new integral, and indicating 
by f a value of t lying between a and ft, we have also 

m^ 



Setting F(t) = 4>() -f- i\lr(), this result may also be written in the 
form 



(3) I~\F(f) ("ffidt, 

Ja 



where X is a complex number whose absolute value is less than or 
equal to unity, this is Darboux's formula. 

To Weierstrass is due a more precise expression, which has a rela- 
tion to some elementary facts of statics When t varies from a to ft, 
the point with the coordinates x = <f> (), y = \fr (t) describes a certain 
curve L Let (X Q , y ), (x v y^ - , (x t _ 19 y^-i); - be the points of 
L which correspond to the values a, t v , ^_ 1? - - of t, and let 



According to a kaown theorem, Z" and F are the coordinates of the 
center of gravity of a system of masses placed at the points (# , y ), 
(x v yj), , (#*_!, ^-0, of the curve X, the mass placed at the 
point (aj^j, y^j) being equal to f(t k ^(t k - t^J, where /(*) is 



66 THE GENERAL CAUGHT THEOBY [n,27 

still supposed to be positive It is clear that the center of gravity 
lies within every closed convex curve C that envelops the curve L 
When the number of mteivals increases indefinitely, the point (X, Y) 
will have for its limit a point whose coordinates (u, v) are given by 
the equations 



~ JLVO* 

which is itself within the curve (7. We can state these two formulae 

as one by wilting 

/ n& 

(4) I = (u 4- w) / /OO dt = Z I /(*) dt, 

Ja <J 

wheie ^ is a point of the complex plane situated with in every closed 
convex cuwe enveloping the curve L It is clear that, in the general 
case, the factoi Z of Weierstrass is limited to a much more lestricted 
region than the factor AJF() of Darboux 

28. Integrals taken along a closed curve. In the preceding para- 
graphs, it suffices to suppose that /(#) is a continuous function of 
the complex variable # along the path of integration We shall now 
suppose also that /() is an analytic function, and we shall first con- 
sidei how the value of the definite integral is affected by the path 
followed by the vanable in going from A to E 

If a function f(z) Is analytic within a dosed curve and also on the 
curve itself y th& integral Jf{&)d& y taken around that curve } w egrual 
to zero 

In order to demonstrate this fundamental theorem, which is due 
to Canchy, we shall first establish several lemmas : 

1) The integrals fdz, Jz dz, taken along any closed curve what- 
ever, are zero In fact, by definition, the integral fda, taken along 
any path whatever between the two points a, b, is equal to b a, 
and the integral is zero if the path is closed, since then I = a. As 
for the integral /# d&, taken along any curve whatever joining two 
points a, I, if we take successively f ft = s^_ l9 then 4 = z k ( 25), 
we see that the integral is also the limit of the sum 

-n(X.f i - O __ yaf+i - ^ _ & - a* 

~ 



hence it is equal to zero if the curve is closed. 

2) If the region bounded by any curve C whatever be divided 
into smaller parts by transversal curves drawn arbitianly, the sum 
of tlie integrals ff(z)dz taken in the same sense along the boundary 



II, 28] 



DEFINITE INTEGRALS 



67 



of each of these parts is equal to the integral //() dz taken along 
the complete boundaiy C. It is clear that each portion of the auxil- 
iary curves sepaiates two contiguous regions and must be described 
twice in integration in opposite senses. Adding all these inte- 
grals, there will remain then only the integrals taken along the 
boundary curve, whose sum is the integial f^f()dz 

Let us now suppose that the region A is divided up, partly in 
smaller regular paits, which shall be squares having their sides 
parallel to the axes Ox, Oy , partly in irregular parts, which shall be 
portions of squares of which the remaining part lies beyond the 
boundaiy C. These squares need not necessarily be equal For ex- 
ample, we might suppose that two sets of parallels to Ox and Oy 
have been drawn, the distance between two neighboimg parallels 
being constant and equal to Z , then some of the squares thus obtained 
might be divided up into smaller squares by new parallels to the 
axes Whatever may be the manner of subdivision adopted, let us 
suppose that there are N regular parts and N 1 irregular parts , let 
us number the regular parts in any order whatever from 1 to A r , and 
the irregular parts from 1 to A T ' Let Z t be the length of the side of 
the zth square and Z that of the square to which the A,th irregular 
part belongs, L the length of the boundary C, and Jl the area of a 
polygon which contains within it the curve C. 

Let abed be the ^th square (Fig. 13), let # t be a point taken in its 
interior or on one of its sides, and let 2 be any point on its boundary, 
Then we have 







FIG. 13 

where |e,| is small, provided that the side of the square is itself 
small. It follows that 

/() = */(*,) +/(*,) - *,/'(*,) +,(*- *,), 



68 THE GENERAL CATJCHY THEORY [II, 28 



=/(*) f 

/<c t ) 



where the integrals are to be taken along the perimetei c t of the 
square By the first lemma stated above, this reduces to the form 

(6) 

Again, let pqrst be the &th irregular part, let z[ be a point taken 
in its interior or on its perimeter, and let z be any point of its 
perimeter Then we have, as above, 



where e* is infinitesimal at the same tune as l' k ; whence we find 
(8) 

Let t] be a positive number greater than the absolute values of 
all the factors c t and e* The absolute value of z s t is less than 
Z l V2 , hence, by (6), we find 



where <u t denotes the area of the tth regular part. From (8) we find, 
in the same way, 

: tfi V2 (4 Zfc + arcrs) = 4 77 VI <* + ^ V2 arcrc, 

where c^ is the area of the square which contains the kfh irregular 
part. Adding all these integrals, we obtain, a fortiori, the inequality 

< t) [4 V2 (Su, + 20 + X V2Z], 

where X is an upper bound for the sides l' k When the number of 
squares is increased indefinitely in such a way that all the sides Z t 
and l' k approach zero, the sum So> t + 54 finally becomes less than Jl 
On the right-hand side of the inequality (9) we have, then, the product 
of a factor which remains finite and another factor 77 which can be 
supposed smaller than any given positive number This can be true 
only if the left-hand side is zero ; we have then 



II, 29] DEFINITE INTEGRALS 69 

29. In order that the preceding conclusion may be legitimate, we must make 
sure that we can take the squares so small that the absolute values of all the 
quantities c t , e \v ill be less than a positive number 17 given in advance, if the 
points Zi and z' k are suitably chosen.* We shall say for brevity that a region 
bounded by a closed curve 7, situated in a region of the plane inclosed by the 
curve C, satisfies the condition (a) with respect to the number 17 if it is possible 
to find in the interior of the curve 7 or on the curve itself a point asf such that 
we always have 

(ex) !/(*) 



when z describes the curve 7 The proof depends on showing that we can choose 
the squares so small that all the parts considered, regular and irregular, satisfy the 
condition (a) with respect to the number ij. 

We shall establish this new lemma by the well-known process of successive 
subdivisions Suppose that we have first drawn two sets of parallels to the axes 
Ox, Oy, the distance between two adjacent parallels being constant and equal 
to I Of the parts obtained, some may satisfy the condition (a), while others 
do not. Without changing the parts which do satisfy the condition (cr), we shall 
divide the others into smaller parts by joining the middle points of the opposite 
sides of the squares which form these parts or which inclose them If, after 
this new operation, there are still parts which do not satisfy the condition (a), 
we will repeat the operation on those parts, and so on Continuing in this way, 
there can be only two cases either we shall end by having only regions which 
satisfy the condition (a), in which case the lemma is proved ; or, however far 
we go in the succession of operations, we shall always find some parts which do 
not satisfy that condition. 

In the latter case, in at least one of the regular or irregular parts obtained 
by the first division, the process of subdivision ]ust described never leads us to 
a set of regions all of which satisfy the condition (a) ; let A 1 be such a part 
After the second subdivision, the part A l contains at least one subdivision -4 2 
which cannot be subdivided into regions all of which satisfy the condition (a) 
Since it is possible to continue this reasoning indefinitely, we shall have a suc- 
cession of regions 

^ii "^2? -^-si * "^ n * " 

which are squares, or portions of squares, such that each is included in the pre- 
ceding, and whose dimensions approach zero as n becomes infinite. There is, 
therefore, a limit point z situated in the interior of the curve or on the curve 
itself Since, by hypothesis, the f unction /(z) possesses a derivative f(z Q ) for 
z = # , we can find a number p such that 



provided that | z z 1 is less than p Let c be the circle with radius p described 
about the point z as center For large enough values of TI, the region A n will 
lie within the circle c, and we shall have for all the points of the boundary of A n 

\fto-f to- <?-**) fW>l^\*-*9\* 

* GOUBSAT, Transactions of the American Mathematical Society t 1900, Vol I, p 14. 



70 THE GENERAL CAUGHT THEORY [II, 29 

Moreover, it is clear that the point Z Q is in the interior of A n or on the boundary , 
hence that region must satisfy the condition (a) with respect to 17 We are 
therefore led to a contradiction in supposing that the lemma is not true 

30. By means of a suitable convention as to the sense of integra- 
tion the theorem can be extended also to boundaries formed by 
several distinct closed curves Let us consider, for example, a func- 
tion /(*) analytic within the region A bounded by the closed curve C 
and the two interior curves C", C", and on these curves themselves 
(Fig 14). The complete boundary T of the region A is formed by 
these three distinct curves, and we shall say that that boundary is 
described in the positive sense if the legion 
A is on the left hand with respect to this 
sense of motion ; the arrows on the figure 
indicate the positive sense of description 
for each of the curves. With this agree- 
ment, we have always 




f 

^ (r) 



- 0, 



p IG 14 the integral being taken along the complete 

boundary in the positive sense. The proof 

given for a region with a simple boundary can be applied again 
here , we can also reduce this case to the preceding by drawing the 
transversals al, cd and by applying the theorem to the closed curve 
abmbandcpcdqa (I, 153). 

It is sometimes convenient m the applications to write the preced- 
ing formula in the form 



= f /(*)&+ r 

J<C"> Jcc 



where the three integrals are now taken in the same sense ; that is, 
the last two must be taken in the reverse direction to that indicated 
by the arrows. 

Let us return to the question proposed at the beginning of 28 ; 
the answer is now very easy Let f(z) be an analytic function in a 
region j4 of the plane. Given two paths AMB 9 ANB, having the same 
extremities and lying entirely in that region, they will give the same 
value for the integral ff(z)d& if the function /() is analytic within 
the closed curve formed by the path AMB followed by the path 
BNA. We shall suppose, for defimteness, that that closed curve 



II, 30] DEFINITE INTEGRALS 71 

does not have any double points Indeed, since the sum of the two 
integrals along ylAfjS and along BXA is zero, the two integials along 
AMB and along ANB must be equal. We can state this lesult again 
as follows Two paths AMB and ANB, having the same extremities, 
give the same value for the integral ff(z)d& if we can pass from one 
to the other by a continuous deformation without encountering any 
point where the function ceases to b& analytic 

This statement holds true even when the two paths have any num- 
ber whatever of common points besides the two extremities (I, 152). 
Fiom this we conclude that, when /(?) is analytic in a region 
bounded by a single closed curve, the integral ff(?)dz is equal to 
zero when taken along any closed curve whatever situated in that 
region But we must not apply this result to the case of a region 
bounded by several distinct closed curves. Let us consider, for exam- 
ple, a function f(z) analytic in the ring-shaped region between two 
concentric circles C, C*. Let C" be a circle having the same center 
and lying between C and C r ; the integral ff(z) dz 9 taken along C", 
is not in geneial zeio. Cauchy's theoiem shows only that the value 
of that integral remains the same when the radius of the cucle C" 
is varied.* 



* Cauehy's theorem remains true without any hypothesis upon the existence of 
the function/ (z) beyond the legion A limited by the cuive C, 01 upon the existence 
of a derivative at each point of the curve C itself It is sufficient that the function/ (z) 
shall be analytic at every point of the region J., and continuous on the boundary C, 
that is, that the value /(Z) of the function in a point Z of C varies continuously with 
the position of Z on that boundary, and that the difference/ (Z) -/(), where z is an 
interior point, approaches zero uniformly with \Z z\ In fact, let us first suppose 
that every straight line from a fixed point a of A meets the boundary in a single 
point When the point z describes C, the point a + 6 (z- a) (where is a real number 
between and 1) describes a closed curve C' situated in A The difference between 
the two integrals, along the curves C and C7', is equal to 



and we can take the difference 1-0 so small that \S] will be less than any given 
positive number, for we can write the function under the integral sign in the foim 



Since the integral along (7 is zero, we have, then, also 



f 

^C 



In the case of a boundary of any form whatever, we can leplace this boundary by a 
succession of closed curves that fulfill the preceding condition by drawing suitably 
placed transversals 



72 THE GENERAL CAUCHY THEORY [II, 31 

31. Generalization of the formulae of the integral calculus. Let/() 
be an analytic function in the region A limited by a simple boundary 
curve C. The definite integral 



taken from a fixed point * up to a variable point ^ along a path 
lying in the region A 9 is, from what we have just seen, a definite 
function of the upper limit Z We shall now show that this function 
*(2T) is also an analytic function of Z whose derivative is f(Z) 
For let ^ + h be a point near - , then we have 



and we may suppose that this last integral is taken along the seg- 
ment of a straight line joining the two points Z and Z + h. If the 
two points are very close together, / (s) differs very little from/(Z) 
along that path, and we can write 

/(*)=/(*) + 8, 

where I SI is less than any given positive number ^ provided that \h\ 
is small enough Hence we have, after dividing by A, 



The absolute value of the last integral is less than iy|&|, and there- 
fore the lefkhand side has for its limit f(Z} when 7i approaches zero 
If a function F(Z) whose derivative is/(Z) is already known, the 
two functions *(2) and P(Z) differ only by a constant (footnote, 
p. 38), and we see that the fundamental formula of integral calculus 
can be extended to the case of complex variables 



(10) A*) ** = *(*i) - F (*o) 

J 

This formula, established by supposing that the two functions f(z), 
F(z) were analytic in the region A, is applicable in more general 
cases. It may happen that the function JP(), or both/(s) and P() 
at the same time, are multiple-valued ; the integral has a precise 
meaning if the path of integration does not pass through any of the 
critical points of these functions. In the application of the formula 
it will be necessary to pick out an initial determination ^( ) of the 
primitive function, and to follow the continuous variation of that 



II, 31] DEFINITE INTEGRALS 73 

function when the variable describes the path of integration. 
Moreover, if f(z) is itself a multiple-valued function, it will be neces- 
sary to choose, among the determinations of F(z), that one whose 
derivative is equal to the determination chosen for/(z). 

Whenever the path of integration can be inclosed within a region 
with a simple boundary, in which the branches of the two functions 
f(z), jF(z) under consideration are analytic, the formula may be 
regarded as demonstrated Now in any case, whatever may be the 
path of integration, we can break it up into several pieces for which 
the preceding condition is satisfied, and apply the formula (10) to 
each of them separately Adding the results, we see that the for- 
mula is true in general, provided that we apply it with the necessary 
precautions. 

Let us, for example, calculate the definite integral *&*&, taken 
along any path whatever not passing through the origin, where t m is 
a real or a complex number different from 1 One primitive func- 
tion is 3 m+1 /(ra + 1), and the general formula (10) gives 



In order to remove the ambiguity present in this formula when m 
is not an integer, let us write it in the form 

Wz = - 



The initial value Log( ) having been chosen, the value of & m is 
thereby fixed along the whole path of integration, as is also the final 
value Log^). The value of the integral depends both upon the 
initial value chosen for Log ( ) and upon the path of integration. 
Similarly, the formula 

<b = Log [/(*,)] - Log [/(* )] 

does not present any difficulty in interpretation if the function f(z) 
is continuous and does not vanish along the path of integration 
The point u =/() describes in its plane an arc of a curve not pass- 
ing through the origin, and the right-hand side is equal to the vari- 
ation of Log(w) along this arc Finally, we may remark in passing 
that the formula for integration by parts, since it is a consequence 
of the formula (10), can be extended to integrals of functions of a 
complex variable 



74 THE GENERAL CAUCHr THEORY [II, 32 

32. Another proof of the preceding results. The properties of the 
integral //()rf* present a gieat analogy to the pioperties of line 
integrals when the condition for mtegrability is fulfilled (I, 152). 
Eiemann has shown, in fact, that Cauchy's theorem results im- 
mediately from the analogous theorem relative to line integrals. 
Let /() = J + Yi be an analytic function of z within a region A 
with a simple boundary , the integral taken along a closed curve C 
lying in that region is the sum of two line integrals . 

/(*)&= f Xcbs-Ydy + i C Ydx + Xdy, 

> J(O J(C) 

and, from the relations which connect the denvatives of the func- 

tions X, Y, ^ __ 1 <?lE __ ? 

dx dy dy dx 

we see that both of these line integrals are zero * (I, 152) 

It follows that the integral f*f(z)dz, taken from a fixed point # 
to a variable point , is a single-valued function <(V) in the region A 
Let us separate the real pait and the coefficient of i in that function . 



/(*, y) /tey) 

P(x, y)s= / Xdx - Ydij, Q(x, y;= I Ydx+Xdy. 

A*o'0o> *A a o'y<r> 

The functions P and Q have partial derivatives, 

J!_ Y ! __ Q_ Q_ 

^r jtL, "T^ J., JC, o^" -^-t 

^a; oy ' ox oy ' 

which satisfy the conditions 

ap_aa ^__^ 

3i ^y Sy "" fix 

Consequently, P + Qi is an analytic function of & whose derivative 



If the function /(#) is discontinuous at a certain number of points 
of A, the same thing will be true of one or more of the functions X l 
7", and the line integrals P(x, y), Q(x, y) will in general have periods 
that arise from loops described about points of discontinuity (I, 153) 
The same thing will then be true of the integral f x z f(z) dz We shall 
resume the study of these periods, after having investigated the nature 
of the singular points of /(). 

* It should be noted that Biemann's proof assumes the continuity of the deriva- 
, dY/dy, , that is, of/'(z) 



It 33] THE CAUCHY INTEGRAL THEOREMS 75 

To give at least one example of this, let us consider the integral f^dz/z. 
After separating the real part and the coefficient of t, we have 

ycte 



r z dz_ __ r (* y)(c + idy __ r <* v>xdx+ ydy 
Ji z J(i, o) x + ly ~~ J(i, o) a 2 + y 2 



(i, o) 



The real part is equal to [log (a 2 4- y 2 )]/2, whatever may be the path followed. 
As for the coefficient of i, we have seen that it has the period 2 ir , it is equal 
to the angle through which the radius vector joining the origin to the point 
(x, y) has turned We thus find again the various determinations of Log(z). 



II CAUCHY'S INTEGRAL TAYLOR'S AND LAURENT'S 
SERIES SINGULAR POINTS RESIDUES 

We shall now present a series of new and important results, which 
Cauchy deduced from the consideration of definite integrals taken 
between imaginary limits. 

33. The fundamental formula. Let/(s) be an analytic function in 
the finite region A limited by a boundary r, composed of one or of 
several distinct closed curves, and continuous on the boundary itself 
If a; is a point * of the region A, the function 



is analytic in the same region, except at the point z = x 

With the point x as center, let us describe a circle y with the 
radius p, lying entirely in the legion A , the preceding function is 
then analytic in the region of the plane limited by the boundary r 
and the circle y, and we can apply to it the geneial theorem ( 28). 
Suppose, for defmiteness, that the boundary r is composed of two 
closed curves (7, ' (Fig. 15) Then we have 



/<*)** ! r /(*) 
*-* JM Z - 



where the three integrals are taken in the sense indicated by the 
arrows. We can write this in the form 



*)fo = r /(*) 
-* ./&>*- 



* In what follows we shall often have to consider several complex quantities at the 
same time We shall denote them indifferently by the letters , z, u, Unless it is 
expressly stated, the letter x will no longer be reserved to denote a real variable. 



76 



THE GENERAL CAUCHY THEORY 



[II, 33 



where the integral L. denotes the integral taken along the total 
boundary T in the positive sense If the radius p of the circle y is 
very small, the value of f(&) at any point of this circle differs very 
hfctle from /(). /()=/(*) + 8, 

where |8| is very small. Replacing /(s) by this value, we find 



The first integral of the right-hand side is easily evaluated , if we 
put z = x 4- pe ei j it becomes 

*" . 

== 2 TTl 

The second integral J^S dz/(z x) is therefore independent of the 
radius p of the circle y, on the other hand, if |S| remains less than 





15 



a positive number 17, the absolute value of this integral is less than 
(y/p) 2 Trp = 2 7Ti7 Now, since the function f(z) is continuous for 
z = x, we can choose the radius p so small that 77 also will be as 
small as we wish Hence this integral must be zero Dividing the 
two sides of the equation (11) by 2 TTI, we obtain 



(12) 



This is Cauchy's fundamental formula It expresses the value of the 
function /(*) at any point x whatever within the boundary by means 
of the values of the same function taken only along that boundary 

Let x + Ace be a point near x, which, for example, we shall suppose 
lies in the interior of the circle y of radius p Then we have also 



II, 33] THE CAUCHY INTEGRAL THEOREMS 77 

and consequently, subtracting the sides of (12) from the correspond- 
ing sides of this equation and dividing by Ace, we find 



/(s)_ 1 r f(z)dz 

Ax 2 m J (r) (z x) (z x Aa) " 

When Ace approaches zero, the function under the integral sign ap- 
proaches the limit f(z)/(z xf In order to prove rigorously that 
we have the right to apply the usual formula for differentiation, let 
us write the integral in the form 



*)<&* , 

2 

Let M be an upper bound for \f(z)\ along r, L the length of the 
boundary, and 8 a lower bound for the distance of any point what- 
ever of the circle y to any point whatever of r The absolute value 
of the last integral is less than ML\&x\/$* and consequently ap- 
proaches zero with |Aa?|. Passing to the limit, we obtain the result 



^QN 
(13) 

It may be shown in the same way that the usual method of differ- 
entiation under the integral sign can be applied to this new integral * 
and to all those which can be deduced from it, and we obtain 
successively 



and, in general, 



& X)" + 1 

Hence, if a function /() is analytic in a certain region of the plane, 
the sequence of successive derivatives of that function is unlimited, 
and all these derivatives are also analytic functions in the same 
region It is to be noticed that we have arrived at this result by 
assuming only the existence of the first derivative. 

Note. The reasoning of this paragraph leads to more general con- 
clusions Let <() be a continuous function (but not necessarily 

* The general formula for differentiation under the integral sign will be established 
later (Chapter V) 



78 THE GENERAL CAUCHY THEORY [II, 33 

analytic) of the complex variable 2 along the curve T, closed or not. 
The integral 



has a definite value for every value of x that does not lie on the 
path of integration. The evaluations just made prove that the limit 
of the quotient [F(x + Aas) JP(a?)]/Aoj is the definite integral 



when |Aoj| approaches zero Hence F(%) is an analytic function for 
every value of x, except for the points of the curve T, which are in 
general singular points for that function (see 90) Similarly, we 
find that the nth derivative F^(x) has for its value 



34. Mbrera's theorem. A converse of Cauchy's fundamental theorem which 
was first proved "by Morera may be stated as follows If a Junction f(z) of a 
complex variable z is continuous in a region A, and if the definite integral f^f(z) dz, 
taken along any closed curve G lying in -4., is zero, then f(z) is an analytic Junc- 
tion in A . 

For the definite integral F(z) = J]/(*)cK, taken between the two points , z 
of the region A along any path whatever lying in that region, has a definite 
value independent of the path If the point Z Q is supposed fixed, the integral 
is a function of z The reasoning of 31 shows that the quotient AF/Az has 
f(z) for its limit when Az approaches zero Hence the function F(z) is an 
analytic function of z having f(z) for its derivative, and that derivative is 
therefore also an analytic function 

35. Taylor's series. Let f(z) be an analytic function in the interior 
of a circle with the center a , the value of that functwn at any point 
x within the circle is equal to the sum of the convergent series 




In the demonstration we can suppose that the function f(z) is 
analytic on the circumference of the circle itself ; in fact, if x is any 
point in the interior of the circle C, we can always find a circle C' 9 
with center a and with a radius less than that of (7, which contains 



II, 35] THE CAUCHY-TAYLOB, SERIES 79 

the point x within it, and we would reason with the circle C 1 just as 
we are about to do with the circle C. With this undei standing, x 
being an interior point of C, we have, by the fundamental formula, 



Let us now write !/( x) in the following way 




or, carrying out the division up to the remamdei of degree n -f- 1 in 
x a, 

1 _ 1 \ x ~ a i OP a) 3 i 

^r^-s-a + ^-^ + ^-a) 8 " 1 " 

+ (a -a)* { (x-aY^ 

Let us replace l/(* a?) in the formula (12^ by this expression, 
and let us bring the factors x a, (x a) 3 , * , independent of z, 
outside of the integral sign. This gives 



where the coefficients J" , / 1? - , / and the remainder R n have the 
values 



(16) 



^ 

n 



As n becomes infinite the remainder R n approaches zero For let 
M be an upper bound for the absolute value of /(*) along the circle 
(7, R the radius of that circle, and r the absolute value of x a. We 
have | * - a? | ^R - r, and therefore 1 1/(* - x) \ ^1/(R - r), when * 
describes the circle C. Hence the absolute value of R n is less than 

1 M _ MR /7-Y+ 1 



and the factor (r/R) n + l approaches zero as n becomes infinite. From 
this it follows that f(x) is equal to the convergent series 



80 THE GENERAL CAUCHY THEORY [II, 35 

Now, if we put x = a in the formulae (12), (13), (14), the boundary 
T being here the circle C, we find 



The series obtained is therefore identical with the series (15) ; that 
is, with Taylor's series 

The circle C is a circle with center a, in the interior of which the 
function is analytic, it is clear that we would obtain the gieatest 
circle satisfying that condition by taking for radius the distance 
from the point a to that singular point of f(z) nearest a This is 
also the circle of convergence for the series on the right * 

This important theorem brings out the identity of the two defini- 
tions for analytic functions which we have given (I, 197, 2d ed , 
191, 1st ed , and II, 3) In fact, every power series represents 
an analytic function inside of its circle of convergence ( 8) , and, 
conversely, as we have ]ust seen, every function analytic in a circle 
with the center a can be developed in a power series proceeding 
according to powers of x a and convergent inside of that circle 
Let us also notice that a certain number of results previously estab- 
lished become now almost intuitive; for example, applying the 
theorem to the functions Log (1 + &) and (1 + #) m , which are ana- 
lytic inside of the circle of unit radius with the origin as center, 
we find again the formulae of 17 and 18 

Let us now consider the quotient of two power series /(#)/< (x), 
each convergent in a circle of radius R. If the series <j>{x) does not 
vanish for x = 0, since it is continuous we can describe a circle of 
radius r 35 R in the whole interior of which it does not vanish The 
f unction /(#)/< (x) is therefore analytic in this circle of radius r and 
can therefore be developed in a power series in the neighborhood 
of the origin (I, 188, 2d ed ; 183, 1st ed ) In the same way, the 
theorem relative to the substitution of one series in another series 
can be proved, etc 

Note Let/(#) be an analytic function in the interior of a circle C 
with the center a and the radius r and continuous on the circle 
itself. The absolute value | f(&) | of the function on the circle is a 
continuous function, the maximum value of which we shall indicate 
by M(r). On the other hand, the coefficient a n of (x a) n in the 

* This last conclusion requires some explanation on the nature of singular points, 
which will be given in the chapter devoted to analytic extension 



II, 37] THE CAUCHY-LAURENT SERIES 81 

development of /() is equal to /<"> ()/', that is, to 



we have, then, 

(17) A n = \a n \< 

so that JXC(r) is greater than all the products A n r** We could use 
3fC(r) instead of M in the expression for the dominant function 
(I, 186, 2d ed , 181, 1st ed ) 

36 Liouville's theorem If the function f(x) is analytic for every 
finite value of x } then Taylor's expansion is valid, whatever a may be, 
in the whole extent of the plane, and the function considered is called 
an ^ntegral function From the expressions obtained for the coeffi- 
cients we easily derive the following proposition, due to Liouville : 

Every integral function whose absolute, value is always less than a 
fixed number M is a constant. 

For let us develop f(x) in powers of x a, and let a n be the 
coefficient of (x a) n It is clear that <%C(r) is less than M, what- 
ever may be the radius r, and therefore \a n \ is less than Jlf/r*. But 
the radius r can be taken just as large as we wish , we have, then, 
a n = if n ^ 1, and f(x) reduces to a constant /(a). 

More generally, let f(x) be an integral function such that the 
absolute value of f(x)/x m remains less than a fixed number M for 
values of x whose absolute value is greater than a positive number 
R ; then thefunctwnf(x) is a polynomial of degree not greater than 
m. For suppose we develop f(x) in powers of x, and let a n be the 
coefficient of x n . If the radius r of the circle C is greater than J?, we 
have JXC(r) < Mr, and consequently |<z n | < Mr-" If n > m, we 
have then a n = 0, since Mr"* can be made smaller than any given 
number by choosing r large enough 

37. Laurent's series. The reasoning by which Cauchy derived 
Taylor's series is capable of extended generalizations. Thus, let 
f(z) be an analytic function in the ring-shaped region between the 

* The inequalities (17) are interesting, especially since they establish a relation 
between the order of magnitude of the coefficients of a power series and the order of 
magnitude of the function, 5W(r) is not, in general, however, the smallest number 
which satisfies these inequalities, as is seen at once when all the coefficients a n are 
real and positive These inequalities (17) can be established without making use of 
Cauchy 's integral (MERAY, Legons nouvelles sur I' analyse irtfimtisfimale, Vol I, p 99). 



82 THE GENERAL CAUCHY THEORY [n, 37 

two concentric circles C, C' having the common center a We shall 
show that the value f(x) of the function at any point x taken in that 
region is equal to the sum of two convergent series, one proceeding in 
positive powers ofx a, the other in positive powers ofl/(x a) * 

We can suppose, just as before, that the function /(#) is analytic 
on the circles f, C' themselves Let E, R 1 be the radii of these circles 
and r the absolute value of x a , if C 1 is the interior circle, we have 
jR' < r < R About x as center let us describe a small circle -y lying 
entirely between C and C' We have the equality 




the integrals being taken in a suitable sense , the last integral, taken 
along y, is equal to 2 7rif(x) } and we can write the preceding relation 
m the form 



(18) /(*)' 

where the integrals are all taken in the same sense 

Repeating the reasoning of 35, we find again that we have 



where the coefficients J 03 J" i; -, J n , are given by the formulae 
(16) In order to develop the second integral in a series, let us 
notice that 



x 




and that the integral of the complementary term, 

i r (i^ym^ 

* m JwV* a ' x ~~* 



approaches zero when n increases indefinitely In fact, if M r is the 
maximum of the absolute value of /(#) along C", the absolute value 
of this integral is less than 



* Comptes rendus de I'Academw des Sciences, Vol XVII See OEuvres de Cauehy, 
1st senes, Vol VHI, p 115 



II, 37] THE CAUCHY-LAURENT SERIES 83 

and the factor R'/r is less than unity. We have, then, also 

(20 ) JL 

^ > 



where the coefficient K n is equal to the definite integral 



Adding the two developments (19) and (20), we obtain the proposed 
development of f(x) 

In the formulae (16) and (21), which give the coefficients / and K n , 
we can take the integrals along any cucle r whatever lying between C 
and C f and having the point a for center, for the functions under the 
integral sign are analytic in the ring. Hence, if we agree to let the 
index n vary from oo to-fao, we can write the development of 
f(x) in the form 

(22) /(*)= + " 

i 

where the coefficient / w , whatever the sign of n, is given by the 
formula 



Example The same f unction /(x) can have developments which are entirely 
different, according to the region considered Let us take, for example, a 
rational fraction /(a;), of which the denominator has only simple roots with 
different absolute values. Let a, &, c, , I toe these roots arranged in the order 
of increasing absolute values Disregarding the integral part, which does not 
interest us here, we have 

+ JL + 



x a a? 6 oj c a I 

In the circle of radius a about the origin as center, each of the simple frac- 
tions can be developed in positive powers of x, and the development off(x) is 
identical with that given by Maclaunn's expansion 



In the ring between the two circles of radii | a\ and |6| the fractions l/(x 6), 
l/(x c), - , l/(x Z) can be developed m positive powers of x, but l/(x a) 
must be developed in positive powers of 1/x, and we have 



84 



THE GENERAL CAUCHY THEORY 



[II, 37 



In the next ring we shall have an analogous development, and so on Finally, 
exterior to the circle of radius |/|, we shall have only positive powers of 1/x 



/(*) = 



+ L Aa,+ 



38. Other series. The proofs of Taylor's series and of Laurent's series are 
based essentially on a particular development of the simple fraction l/(z x) 
when the point x remains inside or outside a fixed circle Appell has shown that 
we can again generalize these formulae by considering a function f(x) analytic 
in the inteiior of a region A bounded by any number whatever of arcs of 




FIG 16 

circles or of entire circumferences * Let us consider, for example, a function 
/(x) analytic in the curvilinear triangle PQR (Fig 16) formed by the three 
arcs of circles PQ, QB, JBP, belonging respectively to the three circumferences 
(7, C", C" Denoting by % any point within this curvilinear triangle, we have 



- 1 C f(z)ds + 1 C /(z)(fe i * C f(e)d 

z) -^ / TTF+R; L, .-_._i L, T= 



Along the arc PQ we can write 
1 1 



x 2 a 



e-a 

~r ~7~ To T ' 



a) 2 



1 /x a' 

>, x \z a, 



where a is the center of C , but when 2 describes the arc PQ, the absolute value 
of (x a)/(z a) is less than unity, and therefore the absolute value of the 
integral 



approaches zero as n becomes infinite. We have, therefore, 



Vol I, p 145 



II, 38] THE CAUCHY-LAUREtfT SERIES 85 

where the coefficients are constants whose expressions it would be easy to write 
out Similarly, along the arc QR we can write 



x-z x- 



~ /-. t\^ ' 



I /z-b\ 

X Z \X &/ 



where 6 is the center of C" Since the absolute value of (z b) n /(x 6) ap- 
proaches zero as n becomes infinite, we can deduce from the preceding equation 
a development for the second integral of tie form 

/m J_ C f&te *! r t Jr. 



Similarly, we find 

j_ 

2 * 

where c is the center of the circle C". Adding the three expressions (a), (), 
(7), we obtain foi/(x) the sum of three semes, proceeding respectively accord- 
ing to positive powers of x - a, of l/(x - 6), and of l/(x c). It is clear that 
we can transform this sum into a series of \rtuch all the terms are rational func- 
tions of a, for example, by uniting all tke terms of the same degree in x a, 
l/(z &), l/(x c) The preceding reasoning applies whatever may be the 
number of arcs of circles 

It is seen in the preceding example that the three series, (a), (0), (7), are 
still convergent when the point x is inside the triangle P'Q'JB', and the sum of 
these three series is again equal to the integral 



taken along the boundary of the triangle PQR m the positive sense. Now, when 
the point x is in the triangle f^Q'J?', th& function f(z)/(z x) is analytic in 
the interior of the triangle PQJS, and the preceding integral is therefore zero 
Hence we obtain in this way a series of ra,tional fractions which is convergent 
when x is within one of the two triangles JPQR, P'Q'.R', and for which the sum 
i& equal tof(x) or to zero, according as thepwnt xisin the triangle PQE or in tTie 
triangle P'QR' 

Pamleve* has obtained more general results along the same lines * Let us con- 
sider, m order to limit ourselves to a very simple case, a convex closed curve T 
having a tangent which changes continuously and a radius of curvature which 
remains under a certain upper bound It is easy to see that we can associate 
with each point M of r a circle C tangent to T at that point and inclosing that 
curve entirely in its interior, and this may be done in such a way that the center 
of the circle moves in a continuous manner with M Let/(z) be a function ana- 
lytic in the interior of the boundary T and continuous on the boundary itself. 
Then, m the fundamental formula 



* Sur les hgnes slngulures desfonctiow malytiques (Annates de laFacultf de 
Toulouse, 1888) 



86 THE GENERAL CAUGHT THEORY [II, 38 

where x is an interior point to r, we can write 

I 1 x-a (x-a) n 1 /x-a\+i 



.+ - +. . + /- :'+()" 

z x z a (z a) 2 (z a) n + - 1 z a? \z a/ 

where a denotes the center of the circle C which corresponds to the point z of 
the boundary , a is no longer constant, as in the case already examined, but it 
is a continuous function of z when the point M describes the curve T - Never- 
theless, the absolute value of (x a)/(z a), which is a continuous function of 
z, remains less than a fixed number p less than unity, since it cannot reach the 
value unity, and therefore the integral of the last term approaches zero as n 
becomes infinite Hence we have 

(25) 



and it is clear that the general term of this series is a polynomial P n (x) of 
degree not greater than n The function f(x) is then developable in a series of 
polynomials in the interior of the boundary T 

The theory of conformal transformations enables us to obtain another kind 
of series for the development of analytic functions Let f(x) be an analytic 
function in the interior of the region J., which may extend to infinity Suppose 
that we know how to represent the region A confoimally on the region inclosed 
by a circle C such that to a point of the region A corresponds one and only 
one point of the circle, and conversely , let u = <f> (z) be the analytic function 
which establishes a correspondence between the region A and the circle C hav- 
ing the point u = for center m the w-plane When the variable u describes 
this circle, the corresponding value of z is an analytic function of u The same 
is true of /(), which can therefore ( be developed in a convergent series of 
powers of u, or of (2), when the variable z remains in the intenor of A 

Suppose, for example, that the region A consists of the infinite strip included 
between the two parallels to the axis of reals y = 0. We have seen ( 22) 
that by putting u = (#*>* !)/(&*&* + 1) this strip is made to correspond to 
a circle of unit radius having its center at the point u = Every function 
analytic in this strip can therefore be developed in this strip in a convergent 
series of the following form 




39. Series of analytic functions. The sum of a uniformly conver- 
gent series whose terms are analytic functions of is a continuous 
function of z, but we could not say without further proof that that 
sum is also an analytic function It must be proved that the sum has 
a unique derivative at every point, and this is easy to do by means 
of Cauchy's integral. 

Let us first notice that a uniformly convergent series whose terms 
are continuous functions of a complex variable # can be integrated 
term by term, as in the case of a real variable The proof given in 



II, 39] THE CAUCHY-LAURENT SERIES 87 

the case of the real variable (I, 114, 2d ed ; 174, 1st ed ) applies 
here without change, provided the path of integration has a finite 
length 

The theorem which we wish to prove is evidently included in the 
following more general proposition 

Let 



be a series all of whose terms are analytic functions in a region A 
bounded by a closed curve F and continuous on the boundary. If the 
series (26) is uniformly convergent on T, it is convergent in every point 
of A, and its sum is an analytic function F(&) whose jt?th derivative 
is represented by the series fonned by the ^>th derivatives of the terms 
of the series (26). 

Let <j> (2) be the sum of (26) in a point of r , < (#) is a continuous 
function of & along the boundary, and we have seen ( 33, Note) 
that the definite integral 

(2T) , (s) 

^ J ^ } 

where x is any point of A, represents an analytic function in the 
region A, whose pfh. derivative is the expression 





(2& 

(28) 

Since the series (26) is uniformly convergent on T, the same thing 
is true of the series obtained by dividing each of its terms by z x, 
and we can write 



or again, since f v () is an analytic function in the interior of r, 
we have, by formula (12), 



Similarly, the expression (28) can be written in the form 



Hence, if the series (26) is uniformly convergent m a region A of 
the plane, x being any point of that region, it suffices to apply the 



88 THE GENERAL CAUCHY THEORY [II, 39 

preceding theorem to a closed curve T lying in A and suuoundmg 
the point x. This leads to the following pioposition 

Every series uniformly convergent in a region A of the plane, whose 
terms are all analytic functions in A, represents an analytic function 
F(z) in the same region The pfh derivative of F(e) is equal to the 
series obtained by differentiating p times each tenn of the series 
which represents F(z) * 

40. Poles. Every function analytic in a circle with the center a is 
equal, in the interior of that circle, to the sum of a power series 

(29) /(*)=^ + ^(*-a) + - +^.(*-a)" + 
We shall say, for brevity, that the function is regular at the point a, 
or that a is an ordinary point for the given function We shall call the 
interior of a circle 0, descubed about a as a center with the radius p, 
the neighborhood of the point a, when the formula (29) is applicable. 
It is, moreover, not necessary that this shall be the largest circle in the 
interior of which the formula (29) is true , the radius p of the neigh- 
borhood will often be defined by some other particular property 

If the first coefficient A Q is zero, we have f(a) = 0, and the point 
a is a zero of the function /() The order of a zero is defined in the 
same way as for polynomials ; if the development of f(z) commences 
with a term of degree m in a } 

/(*) = ^(*-) + ^ + i(*-)" + 1 + ., (m > 0), 
where A m is not zero, we have 

/(a) = 0, /'() = 0, ., /*-() =0, y*->(a)*0, 

and the point a is said to be a zero of order m We can also write 
the preceding formula in the form 



< (&) being a power series which does not vanish when = a Since 
this series is a continuous function of z, we can choose the radius p 
of the neighborhood so small that <(#) does not vanish m that 
neighborhood, and we see that the function /() will not have any 
other zero than the point a in the interior of that neighborhood. 
The zeros of an analytic function are therefore isolated points 

Every point which is not an ordinary point for a single-valued 
f unction f(z) is said to be a singular point A singular point a of the 

* This proposition is usually attributed to Weierstrass 



II, 40] SINGULAR POINTS 89 

f unction /(g) is &pole if that point is an ordinary point for the re- 
ciprocal function !//(). The development of !//() in poweis of 
a cannot contain a constant term, for the point a would then be 
an ordinary point for the function f(z) Let us suppose that the 
development commences with a term of degree m in z a, 

(30) 7^ = (-)-*(*), 

where <() denotes a regular function in the neighboihood of the 
point a which is not zero when z = a. From this we derive 



(31) /(*) = 



where \ff(z) denotes a regular function in the neighboihood of the 
point a which is not zero when z = a. This formula can be written 
in the equivalent form 

(SI 1 ) /(*)=, A \ OT + / Bm \l 1 + - 
x ' ' v y m - 1 



a 



where we denote by P(z a), as we shall often do hereafter, a 
regular function for z = a, and by J3 m , B m _ 19 -, ^ ceitain con- 
stants Some of the coefficients B 19 B^ - , B m _ 1 may be zero, but 
the coefficient B m is surely different from zero. The integer m is 
called the order of the pole It is seen that a pole of order m of f(z) 
is a zero of order m of !//"(), and conversely. 

In the neighborhood of a pole a the development of f(z) is com- 
posed of a regular part P(z a) and of a polynomial in l/(s a)j 
this polynomial is called the principal part of /(#) in the neighbor- 
hood of the pole. When the absolute value of a approaches zero, 
the absolute value off(z) becomes infinite in whatever way ike point 
approaches the pole In fact, since the function ij/(z) is not zero for 
& = a, suppose the radius of the neighborhood so small that the 
absolute value of \j/(z) remains greater than a positive number M in 
this neighborhood. Denoting by r the absolute value of 2 a, we 
have |/()| >l//r m , and therefore |/(s)| becomes infinite when r 
approaches zero. Since the function \fr(z) is regular for z == a, there 
exists a circle C with the center a in the interior of which i/r() is 
analytic. The quotient \j/(z)/(z a) m is an analytic function for all 
the points of this circle except for the point a itself. In the neigh- 
borhood of a pole a, the function /() has therefore no other singulai 
point than the pole itself; in other words, poles are isolated singular 
points* 



90 THE GENERAL CAUCHY THEORY [II, 41 

41 Functions analytic except for poles Every function which is 
analytic at all the points of a legion A, except only for singular 
points that are poles, is said to be analytic except for poles in that 
region* A function analytic in the whole plane except for poles 
may have an infinite number of poles, but it can have only a finite 
number in any finite region of the plane The proof depends on a 
geneial theorem, which we must now recall If in a finite region A 
of the plane there exist an infinite number of points possessing a 
particular propei ty> there exists at least one limit point in the region 
A or on its boundary (We mean by limit point a point in every 
neighborhood of which there exist an infinite numbei of points 
possessing the given propeity) This proposition is pioved by the 
process of successive subdivisions that we have employed so often 
For bievity, let us indicate by (E) the assemblage of points con- 
sidered, and let us suppose that the region A is divided into squares, 
or portions of squares, by parallels to the axes Ox, Oy There will 
be at least one region A 1 containing an infinite number of points of 
the assemblage (E) By subdividing the region A l in the same way, 
and by continuing this process indefinitely, we can form an infinite 
sequence of regions J 1? A 2 , - , A n , that become smaller and 
smaller, each of which is contained in the preceding and contains 
an infinite number of the points of the assemblage. All the points of 
A n approach a limit point Z lying in the interior of or on the bound- 
ary of A. The point Z is necessarily a limit point of (#), since there 
are always an infinite number of points of (.E) in the interior of a 
circle having Z for center, however small the radius of that circle 
may be. 

Let us now suppose that the function f(z) is analytic except for 
poles in the mteiior of a finite legion A and also on the boundary r 
of that region. If it has an infinite number of poles in the region, 
it will have, by the preceding theorem, at least one point Z situated 
in A or on P, in every neighborhood of which it will have an infinite 
number of poles. Hence the point Z can be neither a pole nor an 
ordinary pomt. It is seen in the same way that the function f(z) 
can have only a finite number of zeros in the same region. It follows 
that we can. state the following theorem : 

Every function analytic except for poles in a finite region A and on 
its boundary has in that region only a finite number of zeros and only 
a finite number of poles. 

* Such fanctaoas are said by some writers to be meromorphic. TRANS. 



H, 42] SINGULAR POINTS 91 

In the neighborhood of any point a, a function f(z) analytic 
except for poles can be put in the form 

(32) /(*) = (*-*)**, 

where <f> (z) is a regular function not zero f or z = a The exponent 
fji is called the order of f(z) at the point a The order is zero if the 
point a is neither a pole nor a zero for f(z) , it is equal to m if 
the point a is a zero of oider m for f(z), and to n if a is a pole 
of ordei n for /(). 

42. Essentially singular points. Every singular point of a single- 
valued analytic function, which is not a pole, is called an essen 
tially singular point An essentially singular point a is isolated 
if it is possible to describe about a as a center a circle C in the 
interior of which the function f(z) has no other singular point 
than the point a itself; we shall limit ourselves for the moment 
to such points 

Laurent's theorem furnishes at once a development of the func- 
tion/^) that holds in the neighboihood of an essentially singular 
point Let C be a circle, with the center a, in the interior of which 
the function /(#) has no other singular point than a , also let c be a 
cucle concentric with and ulterior to C. In the circular ring included 
between the two circles C and c the function f(z) is analytic and 
is therefore equal to the sum of a series of positive and negative 
powers of z a, 

(33) /()= 4.(-a)-. 

m= eo 

This development holds true for all the points interior to the circle 
C except the point a, for we can always take the radius of the circle 
c less than \ a\ for any point z whatever that is different from a 
and lies in C f . Moreover, the coefficients A m do not depend on this 
radius ( 37) The development (33) contains first a part regular 
at the point a, say P(z a), formed by the terms with positive 
exponents, and then a series of terms in powers of l/(s a), 



This is the principal part of /(#) in the neighborhood of the singular 
point This principal part does not reduce to a polynomial in 
(& _ #)-i ; for the point z = a would then be a pole, contrary to the 



92 THE GENERAL CAUCHY THEORY [II, 42 

hypothesis * It is an integral transcendental function of l/(z a) 
In fact, let r be any positive number less than the radius of the 
circle C; the coefficient A_ m of the series (34) is given by the 
expression (37) 



the integral being taken along the circle C 1 with the center a and 
the radius r We have, then, 



(35) \A_ m \ 

where M(r) denotes the maximum of the absolute value of /() 
along the circle C' The series is then conveigent, provided that 
| & a \ is greater than r, and since r is a number which we may 
suppose as small as we wish, the series (34) is conveigent for every 
value of & different from a, and we can write 



where P(& a) is a regular function at the point a, and #[!/( a)] 
an integral transcendental functiont of l/(s a) 

When the absolute value of * & approaches zero, the value of 
/() does not approach any definite limit. More precisely, i/ a circle 
C is described with the point a as a center and with an arbitrary 
radius p, there always exists in the interior of this circle points zfor 
which /() differs as little as we please from any number given in 
advance (WBIBBSTBASS) 

Let us first prove that, given any two positive numbers p and M, 
there exist values of z for which both the inequalities, | # a \ < p, 
[/(#)! > M, hold. For, if the absolute value of /(#) weie at most 
equal to M when we have \z a\ < /o, 3fC(r) would be less than 
or equal to M for r < p, and, from the inequality (35), all the coeffi- 
cients A_ m would be zero, for the product c^fT(y)r m ^Mr would 
approach zero with r 

Let us consider now any value A whatever If the equation 
f(z) = A has roots within the circle C, however small the radius p 

* To avoid overlooking any hypothesis, it would he necessary to examine also the 
case m which the development of /(z) in the interior of O contains only positive 
powers of 2- a, the value /(a) of the function at the point a heing different from the 
term independent of z a in the series The point z- a would he a point of dzscorir 
tinuity for/(2) We shall disregard this kind of singularity, which is of an entirely 
artificial character (see helow, Chapter IV). 

f We shall frequently denote an integral function of a; by G(x) 



II, 42] SINGULAR POINTS 93 

may be, the theorem is proved If the equation /(?) = A does not 
have an infinite number of roots in the neighboihood of the point a, 
we can take the radius p so small that in the interior of the circle C 
with the radius p and the center a this equation does not have any 
roots. The function < (2) = l/[/(s) A~\ is then analytic foi every 
point & within C except for the point a ; this point a cannot be any- 
thing but an essentially singular point for <(s), for otherwise the 
point would be either a pole or an ordinary point for/(s). There- 
foie, from what we have just proved, there exist values of z in the 
interior of the circle C for which we have 

|4>(*)|>7 or |/(*) 

however small the positive number may be 

This property sharply distinguishes poles from essentially singu- 
lar points. While the absolute value of the function /(#) becomes 
infinite in the neighborhood of a pole, the value of /() is completely 
indeterminate for an essentially singular point 

Picard * has demonstrated a more precise proposition by showing 
that every equation f(z) = A has an infinite number of roots in the 
neighboihood of an essentially singular point, theie being no excep- 
tion except for, at most, one particular value of A. 

Example The point 2 = is an essentially singular point for the function 



It is easy to prove that the equation e i/z = A has an infinite number of roots 
with absolute values less than />, however small p may be, provided that A is 
not zero Setting A = r (cos B + i sin 0), we deuve from the preceding equation 



z 
We shall have \z\ < p, provided that 



. 

There are evidently an infinite number of values of the integer k which satisfy 
this condition In this example there is one exceptional value of A, that is, 
A 0. But it may also happen that there are no exceptional values , such is 
the case, for example, for the function sm (1/e), near 2 = 

*JLnndle$ de rgcote Normale suptrieure, 1880, 



94 THE GENERAL CAUCHY THEORY [II, 43 

43. Residues. Let a be a pole or an isolated essentially singular 
point of a f unction /(s) Let us consider the question of evaluating 
the integral ff(z) d along the circle C drawn in the neighboihood 
of the point a with the center a The regular part P(& a) gives 
zero in the integration As for the principal part [l/(s a)], we 
can integrate it term by teim, for, even though the point a is an 
essentially singulai point, this series is uniformly convergent The 
integral of the general term 



is zero if the exponent m is greater than unity, for the primitive 
function A_ m /[(m l)(s a)" 1 "" 1 ] takes on again its original 
value after the variable has described a closed path If, on the con- 
trary, w = l, the definite integral A^fdz/fa a) has the value 
2 TrzJLi, as was shown by the previous evaluation made in 34 We 
have then the result ~ 

2iriX_ l = I /(*)<&, 
t/(O 

which is essentially only a particular case of the formula (23) for 
the coefficients of the Laurent development. The coefficient JLi is 
called the residue of the function f(z) with respect to the singular 
point a 

Let us consider now a function /(#) continuous on a closed 
boundary curve r and having m the interior of that curve T only a 
finite number of singular points a, b, c } - , L Let A,B,C, , L be 
the corresponding residues ; if we surround each of these singular 
points with a circle of very small radius, the integral //(#)<#, taken 
along r in the positive sense, is equal to the sum of the integrals 
taken along the small curves in the same sense, and we have the 
very important formula 



(36) C f()<fc=*27ri(A+B+C+ 

Jcr> 



which says that the integral ff(z)dz, taken along T in the positive 
sense, is equal to the product of 2m and the sum of the residues with 
respect to the singular points off(&) within the curve P 

It is clear that the theorem is also applicable to boundaries r com- 
posed of several distinct closed curves. The importance of residues 
is now evident, and it is useful to know how to calculate them rapidly. 
If a point a is a pole of order m for f(z), the product (z a) m f(z) 
is regular at the point <z, and the residue of f(z) is evidently the 



II, 44] APPLICATIONS OF THE GENERAL THEOREMS 95 

coefficient of (2 a) m ~ l in the development of that pioduct. The 
rule "becomes simple in the case of a simple pole ; the residue is then 
equal to the limit of the product (z a)f(z) for * = a. Quite fre- 
quently the f unction f(z) appears under the form 



where the functions P(z) and Q(z) are regular for z = a, and P(a) 
is different from zero, while a is a simple zero for Q(z) Let 
Q(s) = (2 a)R(z), then the residue is equal to the quotient 
P(a)/R (a), or again, as it is easy to show, to P(a)/Q'(a). 

Ill APPLICATIONS OF THE GENERAL THEOREMS 

The applications of the last theorem are innumerable. We shall 
now give some of them which are related particularly to the evalua- 
tion of definite integrals and to the theory of equations. 

44. Introductory remarks. Let f(z) be a function such that the 
product (& a)f(z) approaches zero with \z a\ The integral of 
this function along a circle y, with the center a and the radius p, 
approaches zero with the radius of that circle. Indeed, we can write 



/ 

/Cy) 



If vj is the maximum of the absolute value of (z a)f(z) along the 
circle y, the absolute value of the integral is less than 2 Try, and con- 
sequently approaches zero, since t\ itself is infinitesimal with p. We 
could show in the same way that, when the product (& <j)f(%) 
approaches zero as the absolute value of z a becomes infinite, the 
integral j[ C) /()^j taken along a circle C with the center a, ap- 
proaches zero as the radius of the circle becomes infinite. These 
statements are still true if, instead of integrating along the entire 
circumference, we integrate along only a part of it, provided that 
the product (v a)/(s) approaches zero along that part. 

Frequently we have to find an upper bound for the absolute value 
of a definite integral of the form f a b f(x) dx, taken along the axis of 
reals. Let us suppose for definiteness a < b. We have seen above 
( 25) that the absolute value of that integral is at most equal to the 
integral / a 6 |/(a) | dx, and, consequently, is less than M(b a) HM 
is an upper bound of the absolute value of 



96 THE GENERAL CAUCHY THEORY [II, 45 

45. Evaluation of elementary definite integrals The definite inte- 
gral J**F(x)fa, taken along the real axis, where F(x) is a rational 
function, has a sense, provided that the denominate! does not vanish 
for any real value of x and that the degree of the numeiator is less 
than the degree of the denominator by at least two units. With the 
origin as center let us describe a cncle C with a radius R large 
enough to include all the roots of the denominator of J?(), and let 
us considei a path of integration formed by the diameter A, traced 
along the real axis, and the semicncumference C', lying above the 
real axis. The only singular points of F(z) lying in the interior of 
this path are poles, which come from the roots of the denominator 
of F(%) for which the coefficient of i is positive Indicating by 
3R L the sum of the residues relative to these poles, we can then write 

C F(*)d*+ f F 

JR J(&) 

As the radius R becomes infinite the integral along C" approaches 
zero, since the product *F(*) is zero for * infinite ; and, taking the 

limit, we obtain 

/*+ 

J-co 

We easily reduce to the preceding case the definite integrals 

,*** 

I F(smx,cosx)dx, 
Jo 

where F is a rational function of sin x and cos x that does not 
become infinite for any real value of x } and where the integral is to 
be taken along the axis of reals Let us first notice that we do not 
change the value of this integral by taking for the limits x and 
& + 2 7T 3 where X Q is any real number whatever It follows that we 
can take for the limits TT and + TT, for example Now the classic 
change of variable tan (ar/2) = t reduces the given integral to the 
integral of a rational function of t taken between the limits oo 
and + oo, for tan (x/2) increases from QO to + oo when x increases 
from - TT to + TT 

We can also proceed in another way. By putting e** = & we have 
dx = d/i& 9 and Euler's formulae give 



0080? = ;:; > Sin X = -^-; 

2s 2 is 



II, 46] APPLICATIONS OF THE GENERAL THEOREMS 97 
so that the given integral takes the form 



/ 
J 



2z 



As for the new path of integration, when x increases from to 2 TT 
the vanable # describes in the positive sense the ciicle of unit radius 
about the origin as center It will suffice, then, to calculate the resi- 
dues of the new rational function of z with respect to the poles 
whose absolute values are less than unity 

Let us take for example the integral / 27r ctn [(# a bi)/2]dx, 
which has a finite value if b is not zero. We have 

, fx a bi 
ctn 






/a a fo\ e^-fe- 

( 1* H^^F 




or 

ctn ' ~ ,-. - b + c 
Hence the change of variable e = # leads to the integral 



f *-I-I-T' 

u/(O 



The function to be integrated has two simple poles 

and the corresponding residues are 1 and +2. If b is positive, 
the two poles are in the interior of the path of integration, and the 
integral is equal to 2 iri\ if b is negative, the pole & = is the only 
one within the path, and the integral is equal to 2 m The pro- 
posed integral is therefore equal to 2 iri, according as b is posi- 
tive or negative. We shall now give some examples which are 
less elementary. 

46. Various definite integrals. Example 1 The function & mss /(l + z*) has the 
two poles + 1 and z, with the residues e~ m /2 % and e/^ % Let us suppose 
for defimteness that m is positive, and let us consider the boundary formed by 
a large semicircle of radius R about the origin as center and above the real 
axis, and by the diameter which falls along the axis of reals In the interior of 
this boundary the function e""*/^ + 2 2 ) has the single pole z = $, and the integral 
taken along the total boundary is equal to irer m . Now the integral along the 
semicircle approaches zero as the radius E becomes infinite, for the absolute 
value of the product ze 1 */^ + z*) along that curve approaches zero. Indeed, 
if we replace z by JR (cos 8 + % sin 0), we have 



98 



THE GENERAL CAUCHY THEORY 



[n, 46 



and the absolute value e- 3 ** 1 * 6 remains less than unity when vanes from 
to TT As for the absolute value of the factor z/(l + 2 2 ), it approaches zero as 
z becomes infinite We have, then, m the limit 



'dx = 



x 2 



If we replace e mix by cos mx + * sm mx, the coefficient of % on the left-hand side 
is evidently zeio, for the elements of the integral cancel out in pairs Since we 

have also cos ( mx) = cos mx, we 
can write the preceding formula in 
J& the form 




(37) 



* cos mx , TT 

dx = - 1 

1 + x 2 2 



PIG 17 



with the radii R and.r, and the straight lines AB, B*A' 
We have, then, the relation 



Example 2 The function e z /z is 
analytic in the interior of the bound- 
ary ABMB'A'NA (Fig 17) formed 
by the two semicircles BMB', A'NA, 
described about the origin as center 



-R 



which we can write also in the form 



/Zffix^e-tx r 0* r &z 

X I Z I Z ~~ 

c/(jBJfJB f ) J(A f NA) 

When r approaches zero, the last integial approaches in , we have, in fact, 

&z i 



where P(z) is a regular function at the origin, so that 



r *. 

./(*,' 



* 



The integral of the regular part P (z) becomes infinitesimal with the length of 
the path of integration , as for the last integral, it is equal to the variation of 
Log (z) along A'NA, that is, to m 

The integral along BMB' approaches zero as R becomes infinite Tor if we 
put z = R (cos B + % sin 0), we find 



JL 



OZ = 
, } 2 

and the absolute value of this integral is less than 

r e -.R8iii0tf0 = 2 C 
Jo Jo 



II, 46] APPLICATIONS OF THE GENERAL THEOREMS 99 



When increases from to w/2, the quotient sm 9/0 decreases from 1 to 
2/ir, and we have 



hence 



which establishes the proposition stated above 

Passing to the limit, we have, then (see I, 100, 2d ed.), 



e tar_ e -t 



or 



r 

JQ 

/ -f- 00 

i sma; _ _ v 
Jo ~ ~*' 



Example 3 The integral of the integral transcendental function e- 2 along 
the boundary OABO formed by the two radii OA and OJ5, making an angle of 
45, and by the arc of a circle AB (Fig 18), is 
equal to zero, and this fact can be expressed 
as follows 



C e-^dx + C 

Jo J( 



= C 

J(O 



When the radius E of the circle to which 
the arc AB belongs becomes infinite, the in- 
tegial along the arc AB approaches zero In 
fact, if we put z = E [cos (0/2) + i sm (0/2)], 
that integral becomes 




FIG. 18 



and its absolute value is less than the integral 

i/. 1 --"-** 

As in the previous example, we have 



7? / 

^ f 
2 Jo 



B (-. 
2 Jo 



The last integral has the value 



and approaches zero -when. K becomes infinite. 



100 



THE GENERAL CAUCHY THEORY 



[II, 46 



Along the radius OB we can put z = p [cos(n-/4) + i sm(ir/4)], which gives 
e-* 2 = e-*P 2 , and as E becomes infinite we have at the limit (see I, 135, 2d ed , 
134, 1st ed ) 

r +to */ T ir\, r + ,, VTT 

/ e-*P 2 (cos- + &sm-)(fy> = \ e-^dx^-, 
Jo \ 4 4//o 2 

or, again, 



/ + 2J Vir/ TT TT\ 

I e-*P z dp = - ( cos- - z sin-) 
Jo 2 \ 4 4/ 



Equating the real parts and the coefficients of *, we obtain the values of 
3?resnel's integrals, 



(38) 



47, Evaluation of T (p) 17(1 ^). The definite integral 



Jo 



"*"*/> 1 



where the variable x and the exponent p are real, has a finite value, provided 
that p is positive and less than one , it is equal to the product r (p) T (1 p) * 

In order to evaluate this integral, let 
us consider the function z* ~ l /(l + z), 
which has a pole at the point z = 1 
and a branch point at the point 
z = Let us consider the boundary 
abmb'a'na (Fig 19) formed by the 
two circles. C and C', described about 
the origin with the radii r and p re- 
spectively, and the two straight lines 
ab and a'6', lying as near each other 
as we please above and below a cut 
along the axis Ox The function 
2p-i/(l + z) is single-valued within 
this boundary, which contains only 
19 one singular point, the pole z = 1 

In order to calculate the value of the 

integral along this path, we shall agree to take for the angle of z that one 
which lies between and 2ir If R denotes the residue with respect to the 
pole z = 1, we have then 




The integrals along the circles and C' approach zero as r becomes infinite 
and as p approaches zero respectively, for the product z*/(l + z) approaches 
zero m either case, since <p < 1. 

* Replace t by 1/(1 + a) in the last formula of 135, Vol I, 2d ed , 134, 1st ed 
The formula (39), derived by supposing p to be real, is correct, provided the real part 
ofp lies between and 1 



II, 48] APPLICATIONS OF THE GENERAL THEOREMS 101 

Along ab, 2 is leal For simplicity let us replace 3 by # Since the angle of 
z is zeio along a&, ZP~ I is equal to the numeiical \alue of XP~ I Along afb f 
also z is leal, but since its angle is STT, we have 



The sum of the two integrals along ab and along If of therefore has for its limit 

-de. 



n g2wi(p i)"j i ^ 

Jo 1 + x 



The residue .K is equal to (- l)p-i, that is, to ec^-i)"*, if TT is taken as the 
angle of 1 We have, then, 



1 + a 
or, finally, 
(39) 



/*+ ^-1 ^ v 

Jo 1 + x " ^ 



smjpir 



48. Application to functions analytic except for poles. Given two 
functions, f(z) and < (^), let us suppose that one of them, /(), is 
analytic except for poles in the interior of a closed curve C, that the 
other, <f> (z), is everywhere analytic within the same curve, and that the 
three functions f(z), f r (z), <f> (%) aie continuous on the curve C , and 
let us try to find the smgulai points of the function < (%)/'(%)//(*) 
within C A point a which is neither a pole nor a zero for /(#) is 
evidently an ordinary point for the function f l (&)/f(z) and conse- 
quently for the function $ (#)/ f ()//(#) If a point a is a pole or a 
zeio of(z), we shall have, in the neighborhood of that point, 



where p, denotes a positive or negative integer equal to the order of 
the function at that point ( 41), and where $(%) is a regular func- 
tion which is not zero for & = a Taking the logarithmic derivatives 
on both sides, we find 



Since, on the other hand, we have, in the neighborhood of the point a, 



it follows that the point a is a pole of the first order for the product 
^ (*)/'(*)//(*)> and its residue is equal to f*4(a)> tliat 1S > to w*(a), 
if the point a is a zero of order m for /(*), and to n<f> (a) if the 
point a is a pole of order n for/(s). Hence, by the general theorem 



102 THE GENERAL CAUCHY THEORY [II, 46 

of residues, provided there are no roots of f(z) on the curve O, we 
have 



2^1 f c 



c 

where a is any one of the zeros of f(&) inside the boundary C, I any 
one of the poles of f(z) within C, and where each of the poles and 
zeros are counted a number of times equal to its degiee of multi- 
plicity The formula (40) furnishes an infinite iiumbei of relations, 
since we may take for < (z) any analytic function 

Let us take in particular <j> (z) = 1 , then the preceding formula 
becomes 

(41) *-p 

where N and P denote respectively the number of zeros and the 
number of poles of f(z) within the boundary C This formula leads 
to an important theorem In fact, f'(is)/f(&) is the denvative of 
Log [/(#)] ? to calculate the definite integial on the right-hand side 
of the formula (41) it is therefore sufficient to know the variation of 

log |/(*)| +' angle [/()] 

when the variable # describes the boundary C 111 the positive sense 
But |/() | returns to its initial value, while the angle of /(#) increases 
by 2 -ffTT, K being a positive or negative integer We have, therefore, 



(42) N - P 

that is, the difference N P is equal to the quotient obtained "by the 
division of the variation of the angle off(z) by 2 TT when the variable 
% describes the boundary C in the positive sense 

Let us separate the real part and the coefficient of i *&/(&) 



When the point z = x + yi describes the curve C in the positive- 
sense, the point whose coordinates are X, Y, with respect to a system 
of rectangular axes with the same orientation as the first system, 
describes also a closed curve C f 1 , and we need only draw the curve 
C l approximately in order to deduce from it by simple inspection 
the integer K In fact, it is only necessary to count the number of 
revolutions which the radius vector joining the origin of coordinates 
to the point (X, Y) has turned through in one sense or the other. 



II, 49] APPLICATIONS OF THE GENERAL THEOREMS 103 
We can also write the formula (42) in the form 

/A*\ Ar D 

(43) N-P = 



Since the function F/Z takes on the same value after z has described 
the closed curve C, the definite integral 



/ 

A<?3 



- YdX 



is equal to irI(Y/X\ where the symbol I (Y/X) means the index of 
the quotient Y/X along the boundary C, that is, the excess of the 
number of times that that quotient becomes infinite by passing from 
+ 00 to oo over the number of times that it becomes infinite by 
passing from oo to + oo (I, 79, 154, 2d ed , 77, 154, 1st ed.). 
We can write the formula (43), then, in the equivalent form 

(44) tf-P-| 

49 Application to the theory of equations. When the function f(z) 
is itself analytic within the curve C, and has neither poles nor zeros 
on the curve, the preceding formulae contain only the roots of the 
equation /(#) = which lie within the region bounded by C. The 
formulae (42), (43), and (44) show the number N of these roots by 
means of the variation of the angle of /(#) along the curve or by 
means of the index of Y/X. 

If the function f(z) is a polynomial in 3, with any coefficients 
whatever, and when the boundary C is composed of a finite number 
of segments of umcursal curves, this index can be calculated by ele- 
mentary operations, that is, by multiplications and divisions of 
polynomials. In fact, let AB be an arc of the boundary which can be 
represented by the expressions 



where <j>(t) and \jf(t) are rational functions of a parameter t which 
varies from a to j8 as the point (#, y) describes the arc AB m the 
positive sense. Eeplacing 2 by (<)+ ty(9 m tlie polynomial /(*), 
we have -/ x 



where R (t) and R l (tf) are rational functions of t with real coefficients. 
Hence the index of Y/X along the arc AB is equal to the index of 
the rational function RJR as t vanes from a to ft which we already 



104 THE GENERAL CAUGHT THEORY [n, 49 

know how to calculate (I, 79, 2d ed , 77, 1st ed ) If the bound- 
ary C is composed of segments of unicursal curves, we need only 
' calculate the index for each of these segments and take half of their 
sum, m oider to have the numbei of roots of the equation /() = 
within the boundaiy C 

Note D'Alembeit's theorem is easily deduced from the pieceding 
results Let us piove first a lemma which we shall have occasion to 
use seveial tunes Let F(%) 9 <(#) be two functions analytic in the 
interior of the closed cuive C, continuous on the curve itself, and 
such that along the entire curve C we have |$(s) | <\F(si) \ , under 
these conditions the two equations 

JF(*)=0, F()+*()=0 
have the same number of roots, in the interior of C. For we have 



As the point z describes the boundary C, the point Z = 1 + $ (z)/F(z) 
describes a, closed curve lying entirely within the circle of unit radius 
about the point Z = 1 as center, since | Z 1 1 < 1 along the entire 
curve C Hence the angle of that factor returns to its initial value 
after the variable has described the boundary (7, and the variation 
of the angle of F(z) + <(s) is equal to the variation of the angle of 
F(z) Consequently the two equations have the same number of 
roots in the intenor of C 

'Now let /() be a polynomial of degree m with any coefficients 
whatever, and let us set 



let us choose a positive number R so large that we have 



4. 



*o *" 

Then along the entire circle (7, described about the origin as center 
with a radius greater than R, it is clear that |$/F| < 1 Hence the 
equation f(z) == has the same number of roots in the interior of 
the circle C as the equation F(z) = 0, that is, m. 

50. Jensen's formula Let/(z) be an analytic function except for poles m the 
interior of the circle C with the radius r about the ongin as center, and ana- 
lytic and without zeros on C Let a x , a 2 , - , a* be the zeros, and & 1? & 2 , , b m 
the poles, of f(z) in the interior of this circle, each being counted according to 
its degree of multiplicity We shall suppose, moreover, that the origin is neither 



II, 50] APPLICATIONS OF THE GENERAL THEOREMS 105 
a pole noi a zero for/(a) Let us evaluate the definite integral 
(45) I 



taken along C in the positive sense, supposing that the variable z starts, foi 
example, from the point z = r on the real axis, and that a definite determina- 
tion of the angle of f(z) has heen selected in advance Integrating- by parts, 
we have 



(46) I = {Log (a) Log [/()] } m - J^Log (*) |j cZz, 

where the first part of the right-hand side denotes the increment of the product 
Log (z) Log [f(z)1 when the variable z describes the circle C. If we take zero 
for the initial value of the angle of z, that increment is equal to 

(log r + 2 m) {Log [/(r)] + 2 TTI (n - m)} - log r Log [/(r)] 

= 27Tt Log [/(r)] + 2Tri(n m)logr 4(n- T^Tr 3 



In order to evaluate the new definite integral, let us consider the closed 
curve T, formed by the circumference (7, by the circumference c described 
about the origin with the infinitesimal radius p, and by the two borders a&, 
#?>' of a cut made along the real axis from the point z = p to the point z = r 
(Fig 19) We shall suppose for definiteness that f(z) has neither poles nor 
zeros on that portion of the axis of reals If it has, we need only make a cut 
making an infinitesimal angle with the axis of reals The function Logs is 
analytic in the interior of F, and according to the general formula (40) we 
have the relation 




The integral along the circle c approaches zero with /o ? for the product 
z Log z is infinitesimal with p On the other hand, if the angle of z Is zero 
along ct6, it is equal to 2ir along <&'&', and the sum of the two corresponding 
integrals has for limit 



o f(z) 
The remaining portion is 



and the formula (46) becomes 

(n- m)logr + 2 TTI Log [/(O)] -2mLogs 4(n, - 





In order to integrate along the circle (7, we can put s = r&4 and let <0 vary 
from to 2 ir It follows that dz/z = id<p Let/() = JRe*#, where B and * are 



106 THE GENERAL CAUCHY THEORY [II, 50 

continuous functions of 4> along C Equating the coefficients of i in the preced- 
ing relation, we obtain Jensen's formula * 



(47) _ r log Bfy = log |/(0) | + log 

x ' 



in "which there appear only ordinary Napierian logarithms. 

When the function /(z) is analytic in the interior of 0, it is clear that the 
product ^ - & should be replaced by unity, and the formula becomes 



(48) f "log JB d$ = log |/(0) | + log 

27T JO 

This relation is interesting m that it contains only the absolute values of the 
roots of f(z) within the circle C, and the absolute value of f(z) along that circle 
and for the center of the same circle 

51 . Lagrange's formula. Lagrange's formula, which we have already 
established by Laplace's method (I, 195, 2d ed , 189, 1st ed ), 
can be demonstrated also very easily by means of the general 
theorems of Cauchy. The process which we shall use is due to 
Hermite 

Let /(#) be an analytic function in a certain region D containing 
the point a. The equation ^ ^ 

(49) jr(*)=*-^ + /(*)=0, 

where a is a variable parameter, has the root % a, f or a = 1 Let 
us suppose that a ^ 0, and let C be a circle with the center a and 
the radius r lying entirely in the region D and such that we have 
along the entire circumference o/(s)| < |at a|. By the lemma 
proved in 49 the equation F(z) has the same number of loots 
within the curve C as the equation a = 0, that is, a single loot 
Let f denote that root, and let II (&) be an analytic function in the 
circle C. 

The function U(z)/F() has a single pole m the interior of C, at 
the point 2 = f , and the corresponding residue is n()/F'(). From 
the general theorem we have, then, 



= JL C n(*)<fo = i 

2J<(7) **() 3 *ri 



In order to develop the integral on the right in powers of a, we 
shall proceed exactly as we did to derive the Taylor development, 

* Acta rnatihematica, Vol XXII 

t It is assumed that/(a) is not zero, for otherwise F(z) would vanish when 2 a for 
any value of a and the following developments would not yield any results of 
interest TRANS. 



II, 31] APPLICATIONS OF THE GENERAL THEOREMS 107 
and we shall write 

, 




. 

X* - a)-* 1 * - a - /() * - *J 
Substituting this value in the irrtegial, we find. 



where 

r _ 

~" 




-*-- /() - a 

Let m be the maximum value of the absolute value of of (si) along 
the circumfeience of the circle C, then, by hypothesis, m is less 
than r If M is the maximum value of the absolute value of II (*) 
along C, we have 

which shows that JK n ^ approaches zero when n increases indefinitely. 
Moreover, we have, by the definition of the coefficients J" , /^ - - -, / n , 
. . and the formula (14), 



whence we obtain the following development in series : 



We can write this expression in a somewhat different form If we 
take n ()=/f r () [1 V a/()], "where *(*) is an analytic function in 
the same region, the left-hand side of the equation (50) will no longer 
contain a and will reduce to *() As for the right-hand side, we 
observe that it contains two terms of degree n in a, whose sum is 



{.we/cm- 



108 THE GENERAL CAUCIIY THEORY [II, 51 

\ 

and we find again Lagrange's formula in its usual form (see I, 
formula (52), 195, 2d ed ; 189, 1st ed ) 

O>n 3n 1 

(51) **() + f '()/()+ +^^3lW)[/( a )] w }+ ' - 

We have supposed that we have \af(z)\<r along the cncle C, 
which is true if \a is small enough. In order to find the maximum 
value of | a |, let us limit ourselves to the case where /(*) is a poly- 
nomial or an integral function Let M(r) be the maximum value of 
\f() \ along the cncle C described about the point a as center with the 
radius r The proof will apply to this circle, provided \a\M(r) <r 
We are thus led to seek the maximum value of the quotient r/M(r), 
as r varies from to + oo This quotient is zero for r = 0, for if 
&C(r) weie to approach zero with r, the point z = a would be a zero 
for /(), and F(z) would vanish foi = a. The same quotient is 
also zeio for r = oo, for otherwise /() would be a polynomial of the 
first degree ( 36) Aside from these trivial cases, it follows that 
r/$C(r) passes through a maximum value p for a value r x of r The 
reasoning shows that the equation (49) has one and only one root 
such that | a\<r l9 provided |a|</&. Hence the developments 
(50) and (51) are applicable so long as \a\ does not exceed /*, pro- 
vided the functions H() and &(&) are themselves analytic in the 
circle C t of radius r r 

Example. Let/(z) = (z 2 l)/2 , the equation (49) has the root 



1 Vl 2aa+ a* 
f = - a - ' 

which approaches a when a approaches zero Let us put II (z) = 1. Then the 
formula (50) takes the form 



where X n is the nth Legendre's polynomial (see I, 90, 189, 2d ed , 88, 
184, 1st ed ) In order to find out between what limits the formula is valid, let 
us suppose that a is real and greater than unity On the circle of radius r we 
have evidently #T(r) = [(a + r) 2 1]/2, and we are led to seek the maximum 
value of 2r/[(a + r) 2 1] as r increases from to + > This maximum is 
found for r = Va 2 1, and it is equal to a Va 2 1 If, however, a lies 
between 1 and + 1, we find by a quite elementary calculation that 



The maximum of 2rVl cP/(i* + 1 a 2 ) occurs when r = Vl a 2 , and it is 
equal to unity. 



II, 52] APPLICATIONS OF THE GENERAL THEOREMS 109 



It is easy to verify these results In fact, the ladical Vl-- 2aa+ a: 2 , con- 
sidered as a function of or, has the two critical points a Va 2 1 If a > 1, 
the critical point nearest the origin is a Va 2 1 "When a lies between 1 
and + 1, the absolute value of each of the two critical points a i Vl a 2 is 
unity 

In the fourth lithographed edition of Hermite's lectures will be found (p 185) 
a very complete discussion of Kepler's equation z a = smz by this method 
His piocess leads to the calculation of the root of the transcendental equation 
e r (r 1) = er r (r + 1) which lies between 1 and 2 Stieltjes has obtained the 

values 

fj. = 1 ,199678640257734, p = 6627434193492 

52 Study of functions for infinite values of the variable. In order 
to study a function /() for values of the variable for which the 
absolute value becomes infinite, we can put & = 1/z* and study the 
function /(!/') in the neighborhood of the origin But it is easy to 
avoid this auxiliary transformation We shall suppose first that we 
can find a positive number R such that every finite lalue ofz whose 
absolute value is greater than R is an ordinary point for/(#) If we 
descube a circle C about the origin as center with a radius R, the 
function /(#) will be regular at every point z at a finite distance 
lying outside of C. We shall call the region of the plane exterior 
to C a neighborhood of the point at infinity. 

Let us consider, together with the circle C, a concentiic circle C' 
with a radius R' > R. The function f(&), being analytic in the 
circular ring bounded by C and C f , is equal, by Laurent's theorem, 
to the sum of a series arranged according to integral positive and 
negative powers of 2, 

(53) /(*)= A_ m *; 

7ft= 00 

the coefficients A_ m of this series are independent of the radius R 1 , 
and, since this radius can be taken as large as we wish, it follows 
that the formula (53) is valid for the entire neighborhood of the point 
at infinity, that is, for the whole region exterior to C We shall now 
distinguish several cases : 

1) When the development of f(z) contains only negative powers 
of , 

(54) f^)=^ + A^+A^ 3 +... + A m ^ + ..., 

the f unction /(#) approaches A^ when \z\ becomes infinite, and we 
say that the function f(z) is regular at the point at infinity) or, 
again, that the point at infinity is an ordinary point for f(z). If the 



110 THE GENERAL CAUCHY THEORY [II, 52 

coefficients A Q , A I} , A m _ 1 are zero, but A m is not zeio, the point 
at infinity is a zero of the rath order for f(z). 

2) When the development of f(z) contains a finite number of 
positive powers of z, 

(55) /(*)=*^ + <B-i*"- 1 + 



where the first coefficient B m is not zero, we shall say that the point 
at infinity is a pole of the wth order for /(*?), and the polynomial 
B m z m + - + B^z is the principal part relative to that pole. When 
\z\ becomes infinite, the same thing is tiue of |/(s)|, whatever may 
be the manner in which z moves 

3) Finally, when the development of f(z) contains an infinite 
number of positive powers of #, the point at infinity is an essentially 
singular point for f(&) The series formed by the positive powers of 
z represents an integral function G(z), which is the principal part 
in the neighborhood of the point at infinity. We see in particular 
that an integral transcendental function has the point at infinity as 
an essentially singular point 

The preceding definitions were in a way necessitated by those 
which have already been adopted for a point at a finite distance 
Indeed, if we put z = /z' } the function f(z) changes to a function of 

z', <t> (V)==/(l/*Oj an< ^ ^ 1S seen a ^ once ^ na/ k we k ave on ty carried 
over to the point at infinity the terms adopted for the point z' == 
with respect to the function <f> (#'). Eeasoning by analogy, we might 
be tempted to call the coefficient A_ l of z, in the development (53), 
the residue, but this would be unfortunate In order to preserve the 
characteristic property, we shall say that the residue with respect to 
the point at infinity is the coefficient of l/# with its sign changed, 
that is, A r This number is equal to 



where the integral is taken in the positive sense along the boundary 
of the neighborhood of the point at infinity. But here, the neighbor- 
hood of the point at infinity being the part of the plane exterior to 
C, the corresponding positive sense is that opposite to the usual 
sense. Indeed, this integral reduces to 



II, 32] APPLICATIONS OF THE GENERAL THEOREMS 111 

and, when 2 describes the circle C in the desired sense, the angle of 
diminishes by 2 TT, which gives A as the value of the integral 

It is essential to observe that it is entirely possible for a function 
to be regular at the point at infinity without its residue being zero , 
for example, the function 1 + 1/s has this property. 

If the point at infinity is a pole or a zero for /(), we can write, 
in the neighborhood of that point, 



where ft is a positive or negative integer equal to the order of the 
function with its sign changed, and where < () is a function which 
is regulai at the point at infinity and which is not zero for = oo. 
Fiom the preceding equation we deduce 



where the function <p'(z)/<l>(z) is regular at the point at infinity but 
has a development commencing with a term of the second or a higher 
degiee in 1/s. The residue of f(x)/f(z) is then equal to /*, that 
is, to the older of the f unction /(s) at the point at infinity. The state- 
ment is the same as for a pole or a zero at a finite distance. 

Let/(#) be a single-valued analytic function having only a finite 
number of singular points. The convention which has just been 
made for the point at infinity enables us to state in a very simple 
form the following general theorem 

The sum of the residues of the function f(z) in the entire plane, 
the point at infinity included, is &ero. 

The demonstration is immediate Describe with the origin as 
center a circle C containing all the singular points of /(*) (except 
the point at infinity) The integral //() dz, taken along this circle 
in the oidinary sense, is equal to the product of 2 iri and the sum 
of the residues with respect to all the singular points of f(z) at a 
finite distance On the other hand, the same integral, taken along 
the same circle in the opposite sense, is equal to the product of 2 iri 
and the residue relative to the point at infinity. The sum of the two 
integrals being zero, the same is true of the sum of the residues. 

Cauchy applied the term total residue (residu integral) of a func- 
tion /(#) to the sum of the residues of that function for all the 
singular points at a finite distance. When there are only a finite 
number of singular points, we see that the total residue is equal to 
the residue relative to the point at infmity with its sign changed. 



112 THE GENERAL CAUCHY THEORY [II, 52 

Example. Let 
* 



where P( and Q() are two polynomials, the first of degree p, the 
second of even degree 2q. If 22 is a real number greater than the 
absolute value of any root of Q(, the function is single-valued out- 
side of a circle C of radius R, and we can wxite 



where <(>) is a function which is regular at infinity, and which is 
not zero for * = oo. The point at infinity is a pole for/() iS.p>q 9 
and an ordinary point ifp&q. The residue will certainly be zero 
if p is less than q 1. 

IV. PERIODS OF DEFINITE INTEGRALS 

53* Polar periods. The study of line integrals revealed to us that 
such integrals possess periods under certain circumstances Since 
every integral of a function /(*) of a complex variable * is a sum of 
line mtegials, it is clear that these integrals also may have certain 
periods. Let us consider first an analytic function /(*) that has only 
a finite number of isolated singular points, poles, or essentially 
singular points, within a closed curve C. This case is absolutely 
analogous to the one which we studied for line integrals (I, 153), 
and the reasoning applies here without modification Any path that 
can be drawn within the boundary C between the two points & , Z 
of that region, and not passing through any of the singular points 
of /(*), is equivalent to one fixed path joining these two points, 
preceded by a succession of loops starting from z^ and surrounding 
one or more of the singular points a v a^ * , a n of /() Let A v A^ 
, A+ be the corresponding residues of /(*) ; the integral //(*)<&, 
taken along the loop surrounding the point a l9 is equal to 2 mA l9 
and similarly for the others. The different values of the integral 
fff(*)d& are therefore included in the expression 

r* 

(56) I /(*) d = F(Z) + 2 Tri (m l A l + m 2 A 2 + + m n A n ), 

J** 

where F(Z) is one of the values of that integral corresponding to 
the determined path, and m t , m 2 , - are arbitrary positive or nega- 
tive integers , the periods are 



n, 53] PERIODS OF DEFINITE INTEGRALS 113 

In most cases the points a v a^ , a n are poles, and the periods 
result from infinitely small ciicuits described about these poles , 
whence the teim polar periods, which is oidmarily used to distin- 
guish them from peiiods of another kind mentioned later. 

Instead of a region of the plane interior to a closed curve, we may 
consider a portion of the plane extending to infinity , the function 
f(z) can then have an infinite number of poles, and the integral an 
infinite number of peiiods. If the residue with respect to a singu- 
lar pomt a of f(z) is zero, the corresponding period is zero and the 
point a is also a pole or an essentially singular point for the integral. 
But if the residue is not zero, the point a is a logarithmic critical 
point for the integral. If, for example, the point a is a pole of the 
mth order for/(s), we have in the neighborhood of that point 



and therefore 

R 

. +3 1 Log( a) 




where C is a constant that depends on the lower limit of integration 
* and on the path followed by the variable in integration 

When we apply these general considerations to rational functions, 
many well-known results are at once apparent. Thus, in order that 
the integral of a rational function may be itself a rational function, 
it is necessary that that integral shall not have any periods ; that is, 
all its residues must be zero. That condition is, moreover, sufficient 
The definite integral C z d 



has a single critical point z = a, and the corresponding period is 
2 Tri , it is, then, in the integral calculus that the true origin of the 
multiple values of Log(s a) is to be found, as we have already 
pointed out in detail in the case of ffdz/z ( 31). 
Let us take, in the same way, the definite integral 



dz 



r C 

Jo 

it has the two logarithmic critical points + i and i, but it has only 
the single period TT. If we limit ourselves to real values of the 



114 THE GENERAL CAUCHY THEORY [II, 53 

variable, the diffeient determinations of arc tana; appear as so many 
distinct functions of the vanable x We see, on the contrary, how 
Cauehy's woik leads us to regard them as so many distinct branches 
of the same analytic function 

Note "When there aie more than three periods, the value of the definite 
integral at any point z may be entirely indeterminate Let us recall first the 
following result, taken from the theory of continued fractions* Given a real 
irrational numbei or, we can always find two integers p and <?, positive or nega- 
tive, such that we have \p + qa\ < e, where e is an arbitianly preassigned 
positive number 

The numbers p and q having been selected in this way, let us suppose that 
the sequence of multiples of p + qa is formed Any real number A is equal to 
one of these multiples, or lies between two consecutive multiples We can 
therefore find two integers m and n such that | m + no: A\ shall be less than e, 

With this in mind, let us now consider the function 

2 m \2 a zb z c z d)' 

where a, &, c, d are four distinct poles and or, /3 are leal irrational numbers 
The integral f z *f(z)dz has the four periods 1, <z, i, ifi. Let I(z) be the value of 
the integral taken along a particular path from z to z, and let M + Ni denote 
any complex number whatevei We can always find f oui integers m, n, m', n' 
such that the absolute value of the diffeience 



na + %(mf + n'p) 

will be less than any preassigned positive number e We need only choose 
these integers so that 



, 
2 

where M + Ni I(z) A + Bi Hence we can make the variable describe a 
path joining the two points given in advance, 2 , 2, so that the value of the inte- 
gral ff(z) dz taken along this path differs as little as we wish from any pre- 
assigned number Thus we see again the decisive influence of the path followed 
by the variable on the final value of an analytic function 

54. A study of the integral f*dz/-Vl z 2 The integral calculus 
explains the multiple values of the function arc sin & in the simplest 
manner by the preceding method They arise from the different 
determinations of the definite integral 



according to the path followed by the variable. For defimteness we 
shall suppose that we start from the origin with the initial value + 1 

* A little farther on a direct proof will be f ound ( 66) 



II, 54] PERIODS OF DEFINITE INTEGRALS 115 

for the radical, and we shall indicate by I the value of the integral 
taken along a determined path (01 direct path) Foi example, the 
path shall be along a straight line if the point # is not situated on 
the real axis 01 if it lies upon the real axis within the segment from 
1 to + 1 , but when z is real and \\ > 1, we shall take for the 
duect path a path lying above the real axis 

Now, the points =+l,s= 1 being the only critical points of 
Vl 3 , every path leading fiom the origin to the point can be 
xeplaced by a succession of loops described about the two critical 
points 4- 1 and 1, followed by the direct path. We are then led 
to study the value of the 
mtegial along a loop. Let 
us considei, for example, 
the loop OamaO, described 
about the point & = + 1 ; 
this loop is composed of the segment Oa passing from the origin to 
the point 1 c, of the cucle ama of radius c described about # = 1 
as center, and of the segment aO Hence the integral along the loop 
is equal to the sum of the integials 

dx 





The integral along the small circle approaches zero with c, for the 
product (z 1) f(z) approaches zero. On the other hand, when z 
has described this small circle, the radical has changed sign and in 
the integral along the segment aO the negative value should be 
taken for Vl x 2 The integial along the loop is therefore equal to 
the limit of 2f Q l ~ e dx/^/l a? as e approaches zero, that is, to TT 
It should be observed that the value of this integral does not depend 
on the sense in which the loop is described, but we return to the 
origin with the value 1 for the radical 

If we were to describe the same loop around the point *? = + 1 
with 1 as the initial value of the radical, the value of the integral 
along the loop would be equal to TT, and we should return to the 
origin with + 1 as the value of the radical. In the same way it is 
seen that a loop described around the critical point # = 1 gives 

TT or + TT f or the integral, according as the initial value + 1 or 

1 is taken for the radical on starting from the origin. 

If we let the variable describe two loops in succession, we return 
to the origin with + 1 for the final value of the radical, and the 
value of the integral taken along these two loops will be + 2 TT, 0, or 



116 THE GENERAL CAUCHY THEORY [n, 54 

2 TT, according to the oider in which these two loops are descubed 
An even nuinbei of loops will give, then, 2 mir foi the value of the 
integral, and will bung back the ladical to its initial value +1 
An odd number of loops will give, on the contiary, the value (2 m -f 1) TT 
to the integial, and the final value of the ladical at the origin will 
be 1 It follows from this that the value of the integial F(z) will 
be one of the two forms 



according as the path described by the variable can be replaced by 
the dnect path preceded by an even number or by an odd number 
of loops 

55. Periods of hyperelliptic integrals. We can study, in a similar 
manner, the different values of the definite integral 



(58) 

where P (&) and R (#) are two polynomials, of which the second, R (& 
of degree n, vanishes for n distinct values of z 




We shall suppose that the point # is distinct from the points e v e 2 , 
, e n ; then the equation w 2 = R (Z Q ) has two distinct roots + and 
U Q We shall select w for the initial value of the radical R () If 
we let the variable & descube a path of any form whatever not pass- 
ing through any of the critical points e^ e 2 , , e n , the value of the 
radical V.R() at each point of the path will be determined by con- 
tinuity Let us suppose that from each of the points e l9 e 2 , , e n 
we make an infinite cnt in the plane in such a way that these cuts do 
not cross each other The integral, taken from up to any point z 
along a path that does not cross any of these cuts (which we shall 
call a direct path), has a completely determined value I(z) for each 
point of the plane. We have now to study the influence of a loop, 
described from around any one of the critical points e t , on the 
value of the integral Let 2 E % be the value of the integral taken 
along a closed curve that starts from and incloses the single criti- 
cal point e v the initial value of the ladical being w . The value of 
this integral does not depend on the sense in which the curve is 
described, but only on the initial value of the radical at the point . 
In fact, let us call 2 E{ the value of the integral taken along the same 



II, 35] 



PERIODS OF DEFINITE INTEGRALS 



117 



curve in the opposite sense, with the same initial value II Q of the 
radical If we let the variable # describe the curve twice in succes- 
sion and in the opposite senses, it is clear that the sum of the inte- 
grals obtained is zero ; but the value of the integial for the first turn 
is 2 E l} and we return to the point 2 with the value U Q for the radi- 
cal The integral along the curve described in the opposite sense is 
then equal to 2 E^ 9 and consequently E( = E % . The closed curve 
considered may be reduced to a loop formed by the straight line #, 
the circle c % of infinitesimal radius about e l} and the straight line az Q 
(Fig. 21) , the integral along c t is infinitesimal, since the pioduct 
(# e t ) P (z)/-*jR(z) approaches zero with the absolute value of n 0,. 
If we add together the integrals 
along z Q a and along az Q , we find 





21 



where the integral is taken along 
the straight line and the initial 
value of the radical is 

This being the case, the inte- 
gral taken along a path which 
reduces to a succession of two 
loops described about the points 
e a , e$ is equal to 2 J2* 2 Ep 9 
for we return after the first loop 
to the point # with the value 

U Q for the radical, and the integral along the second loop is equal 
to 2Ep After having described this new loop we return to the 
point # with the original initial value U Q If the path described by 
the variable # can be reduced to an even number of loops described 
about the points e a , e& e y , e S) - -, e K , e K successively, followed by the 
direct path from # to #, where the indices a, ft, , K, \ are taken 
from among the numbers 1, 2, , n, the value of the integral along 
the path is, by what precedes, 

+2(E K - EJ. 



JF(*) = 1+ 2(E a - J 

If, on the contrary, the path followed by the variable can be reduced 
to an odd number of loops described successively around the critical 
points e a , e^ , e K , e^, e^ the value of the integral is 



118 THE GENERAL CAUCHY THEOKY [II, 55 

Hence the integral under consideiation has as periods all the expres- 
sions 2(E, - E h ), but all these periods reduce to (n 1) of them 



for it is clear that we can mite 
2(E t - E h }=2(E t - E 

Since, on the othei hand, 2 E^ = v + 2 E n , we see that all the values 
of the definite integral F(z) at the point are given by the two 
expressions 



where m i; w 2 , , w n _! are arbitiary integers 

This lesult gives use to a certain number of impoitant observa- 
tions. It is almost self-evident that the penods must be independent 
of the point 3 chosen for the starting point, and it is easy to verify 
this Considei, for example, the penod 2E % ~-~2 E h ; this period is 
equal to the value of the integral taken along a closed curve r pass- 
ing through the point Z Q and containing only the two cutical points 
e l} e h . If, for defimteness, we suppose that there are no other critical 
points in the interior of the triangle whose vertices are # , e %9 e h , this 
closed curve can be replaced by the boundary bb'nc'emb (Fig 21) ; 
whence, making the radii of the two small circles approach zero, we 
see that the period is equal to twice the integral 




taken along the straight line joining the two critical points t , e h . 

It may happen that the (n 1) periods t^, w 2 , - , (o n-1 are not 
independent This occurs whenever the polynomial R (2) is of even 
degree, provided that the degree of P(z) is less than n/2 1. With 
the point * as center let us draw a circle C with a radius so large 
that the circle contains all the critical points , and for simplicity let 
us suppose that the critical points have been numbered from 1 to n 
in the order in which they are encountered by a radius vector as it 
turns about in the positive sense 

The integral 




taken along the closed boundary ^AMA^ formed by the radius # Q A, 
by the circle (7, and by the radius Az Q described in the negative sense, 



IU55] PERIODS OF DEFINITE INTEGRALS 119 

is zero The integrals along %^A and along Az Q cancel, for the circle 
C contains an even number of critical points, and after having 
described this circle we return to the point A with the same value 
of the radical On the other hand, the integral along C approaches 
zero as the radius becomes infinite, since the product zP(z)/^R(z) 
approaches zero by the hypothesis made on the degree of the poly- 
nomial P(z) Since the value of this integral does not depend on the 
radius of C, it follows that that value must be zero. 

Now the boundaiy z Q AMAz Q considered above can be replaced by 
a succession of loops described around the critical points e l9 e^ * - - , e n 
in the order of these indices Hence we have the relation 



which can be written in the form 

! <* 2 + <* <4+ + <-! ==0; 

and we see that the n 1 periods of the integral reduce to n 2 
peuods oij, 2 , , o) n _ 2 

Consider now the more general form of integral 



= 

Jz n 



where P, Q, E are three polynomials of which the last, E(z), has only simple 
roots. Among the roots of Q (z) there may be some that belong to E (z) , let or 1T 
o: 2 , , <x g be the roots of Q (z) which do not cause E (z) to vanish. The integral 
F(z) has, as above, the periods 2(J t -Z^), where 2 E t denotes always the inte- 
gral taken along a closed curve starting from z and inclosing none of the roots 
of either of the polynomials Q(z) and E(z) except e, But F(z) has also a cer- 
tain number of polar periods arising from the loops described about the poles 
<ar p (3r a , -, a t The total number of these periods is again diminished by unity 
if E (z) is of even degree n, and if 



where p and q are the degrees of the polynomials P and Q respectively 

Example Let E (z) be a polynomial of the fourth degree having a multiple 
root Let us find the number of periods of the integral 




If E (z) has a double root e l and two simple roots e 2 , %, the integral 

dz 



120 THE GENERAL CAUCHY THEORY [il, 55 

has the period 2 E z 2J 3 , and also a polar penod arising fiom a loop aiound 
the pole e l By the remark made ]ust above, these two periods are equal If 
E (z) has two double loots, it is seen immediately that the integral has a single 
polar period 

If E (z) has a tuple root, the integral 

dz 



has the period 2 jE^ 2 J 2 , but, by the general remaik made above, that period 
is zeio The same thing is tiue if E(z) has a quadiuple loot In resume' we 
have If E (z) has one or two double roots, the integi al has a pet lod , if E (z) has a 
tnple or quadruple root, the integral does not have periods All these results are 
easily venfied by direct integration 

56. Periods of elliptic integrals of the first kind. The elliptic integral 
of the first kind, 




where E (z) is a polynomial of the third or the fourth degree, prime to 
its derivative, has two periods by the preceding general theory We 
shall now show that the ratio of these two periods ^s not real 

We can suppose without loss of generality that R() is of the 
third degree Indeed, if R^z) is a polynomial of the fourth degree, 
and if a is a root of this polynomial, we may write (I, 105, note, 
2ded ; 110, 1st ed) 




where & = a + 1/y and where R (y) is a polynomial of the third 
degree It is evident that the two integrals have the same periods 
If E(z) is of the third degree, we may suppose that it has the loots 
and 1, for we need only make a linear substitution = a + py to 
reduce any other case to this one Hence the proof reduces to 
showing that the integral 

(59) F(z) 

' W 



, 
V*(l -*)(*-*) 

where a is different from zero and from unity, has two periods whose 
ratio is not real 

If a is real, the property is evident Thus, if a is greater than 
unity, for example, the integral has the two periods 



c i ** 2 

Jo Vs(l-3)(a-3)' J 



II, 56] 



PERIODS OF DEFINITE INTEGRALS 



121 



of which the first is real, while the second is a pure imaginary. 
Moreover, none of these periods can be zero. 

Suppose now that a is complex, and, for example, that the coeffi- 
cient of i in a is positive. We can again take for one of the penods 

dz 



We shall apply Weierstrass's formula ( 27) to this integral. When 

* varies from to 1, the factor l/Vg(l z] remains positive, and 

the point representing I/ V# z describes a curve L whose general 

nature is easily determined. Let A 

be the point representing a ; when 

z varies from to 1, the point a & 

descubes the segment AB parallel 

to Ox and of unit length (Fig 22). 

Let Op and Oq be the bisectors of 

the angles which the straight lines 

OA and OB make with Ox, and let 

Op 1 and Oq 1 be straight lines sym- 

metrical to them with respect to Ox. 

If we select that determination of 

Va z whose angle lies between 

and 7T/2, the point V& z de- 



FIG 22 



scribes an are aft from a point a on Op to a point /3 on Oq , hence the 
point I/ V# 3 describes an arc #'' from a point #' on Op 1 to a point 
ft' of 0#'. It follows that Weierstrass's formula gives 





where ^ is the complex number corresponding to a point situated in 
the interior of every convex closed curve containing the aic a 1 ft 1 . It 
is clear that this point Z l is situated in the angle p'Og', and that it 
cannot be the origin , hence the angle of Z^ lies between 7r/2 and 0. 
We can take for the second period 



d* o C 

= 9, I 

V*(l-*)(a-*) Jo 



or, setting z = atf, 



2 



122 THE GENERAL CAUCHY THEORY [II, 56 

In order to apply Weierstrass's formula to this integral, let us notice 
that as t increases from to 1 the point at describes the segment 
OA and the point 1 at describes the equal and parallel segment 
from s = 1 to the point C Choosing suitably the value of the 
radical, we see, as before, that we may write 



/*(! - 

where Z^ is a complex number different from zero wJwse angle lies 
between and -rr/2 The latio of the two periods ^/fy or 2jZ 1 is 
therefore not a real number. 



EXERCISES 
1, Develop the function 



in powers of , m being any number 

Emd the radius of the circle of convergence 

2 Find the different developments of the function l/[(z 3 + 1) (z 2)] in posi- 
tive or negative powers of 2, according to the position of the point z in the plane 

3 Calculate the definite integral /2 2 Log[(z + l)/(z - l)]dz taken along a 
circle of radius 2 about the origin as center, the initial value of the logarithm at 
the point 2 = 2 being taken as real 

Calculate the definite integral 

dz 



taken over the same boundary 

4 Let f(z) be an analytic function in the interior of a closed curve C con- 
taining the origin Calculate the definite integral j^f^LogzcZs, taken along 
the curve C, starting with an initial value z 

6 Derive the relation 



f 

+j 



dt __ 135* (2n-l) 
i~ 246 -2w 



and deduce from it the definite integrals 



r 

y oo 



6 . Calculate the following definite integrals by means of the theory of residues . 

-^ ~-i m and a being real, 

c($ 2 -h a 2 ) 2 



r-j 

Jo x 

r 

/ 00 



cos OKB , , . 

-daj, a being real, 



n,Exs] EXERCISES 123 

+ * 



L 

r 



, a and p being real, 

cosxdx 

"JT J 

+ ao jclogg(%c 



r 

Jo 



(l + <") 8 Jo 

3 cos ax cos & 



dte, a and 6 being real and positive 



(To evaluate the last integral, integrate the function (eP lz Pw)/s 2 along the 
boundary indicated by Fig 17 ) 

7 The definite integral f Q v d<p/[A + C ( J. C) cos 0] is equal, when it 
has any finite value, to eir/^/AC, where e is equal to 1 and is chosen in 
such a way that the coeflicient of t in ei VAC /A is positive 

8 Let 2^(2) and (z) be two analytic functions, and 2 = a a double root of 
G(z) = that is not a root of F(z) Show that the corresponding residue of 
F(z)/G(z) is equal to 

6 ff'(q) G"(a) - 2 F(a) G'"(a) 



In a similar manner show that the residue of F(z)/[G(z)f for a simple root 
a of G (z) = is equal to 

F'(a)G'(a)-F(a)G"(a) 

[G'(a)]* 
9 Derive the formula 



the integral being taken along the real axis with the positive value of the 
radical, and a being a complex number or a real number whose absolute value 
is greater than unity. Determine the value that should be taken for Vl a 2 

10 Consider the integrals J^cfe/Vl + 2 8 , J^dz/Vl + z*, where S and S^ 
denote two boundaries formed as follows The boundary S is composed of a 
straight-line segment OA on Ox (\vhich is made to expand indefinitely), of the 
circle of radius OA about as center, and finally of the straight line AO. The 
boundary S l is the succession of three loops which inclose the points a, 6, c 
which represent the roots of the equation z 5 + 1 = 0. 

Establish the relation that exists between the two integrals 



/1 + x* 
which arise in the course of the preceding consideration 

11 By integrating the function e-* 8 along the boundary of the rectangle 
formed by the straight lines y = 0, y = 6, x = + E, x = 12, and then making 
R become infinite, establish the relation 

er *? cos 2 bx dx = 



124 THE GENERAL CAUCHY THEORY [II,Exs 

12. Integrate the function tr z &~\ where n is real and positive, along a 
boundary formed by a ladius OA placed along Ox, by an aic of a circle AB of 
radius OA about as center, and by a radius SO such that the angle a. = AOB 
lies between and w/2. Making OA become infinite, deduce from the preced- 
ing the values of the definite integrals 

C w w n-i e -att COS & u d f ^-^e-^sin&ttdw, 
Jo Jo 

where a and b are real and positive The results obtained are valid for a = ?r/2, 
provided that we have n < 1 

13 Let m, m', 71 be positive integers (m < w, ra' < n) Establish the formula 



- T /2m +1 \ . /2m" + 1 VI 

t = ctnl TT ctn( ir] 

2nL \ 2n / \ 2n ] 



14 Deduce from the preceding result Euler's formula 



/' 

Jo 



15. If the real part of a is positive and less than unity, we have 



f 

*/ 



1 + & sin air 



(This can be deduced from the formula (39) ( 47) or by integrating the 
function e as /(l + c a ) along the boundary of the rectangle formed by the straight 
lines ?/ = 0, 2/ = 2?r, = +E, x= JB, and then making E become infinite ) 

16 Derive in the same way the relation 







-dec = TT (ctn aw ctn &?r), 



where the real parts of a and 6 are positive and less than unity 

(Take for the path of integration the rectangle formed by the stiaight lines 

y = 0, y = TT, x = E, x = 12, and make use of the preceding exercise ) 
17 Erom the formula 



C 

J((7 



where n and & are positive integers, and G is a circle having the origin as 
center, deduce the relations 



( 2 co SW )+icos( TO - *)<! = ,( + !)( + ) ( + *) 



/ 



+ 1 x* n dx 



1 3 6 (2n-l) 



,_ x /1-JZJ 2 246 2n 

(Put 2 = e 2w , then cos w = $, and replace n by n + fc, and A; by n.) 
18*. The definite integral 



II, Exs ] EXERCISES 125 



when it has a finite value, is equal to TT/ V 1 2 <xx -f a: 2 , where the sign 
depends upon the relative positions of the two points a and x Deduce from 
this the expression, due to Jacobi, for the nth Legendre's polynomial, 



= - C 

If /0 



19. Study in the same way the definite mtegial 



f 

Jo 



J x a + \ x 2 1 cos ^ 
and deduce from the result Laplace's formula 



o 

where e = 1, according as the real part of x is positive or negative 
20* Establish the last result by integrating the function 

1 



along a circle about the origin as center, -whose radius is made to become infinite. 
21* Gauss's sums. Let T s e**/*, where n and s are integers , and let 
S n denote the sum T Q + T : + + T n _i Deuve the formula 



2 

(Apply the theorem on residues to the function &(z) = e 2msZ/n /(e' 2 ' tnz 1), talang 
for the boundary of integration the sides of the rectangle formed by the straight 
lines x = 0, x = n, y = + JS, y = E, and inserting two semicircumf erences of 
radius e about the points x = 0, x = n as centers, in order to avoid the poles 
2 = and z = n of the function <j> (z) , then let It become infinite ) 

22 Let/(z) be an analytic function in the interior of a closed curve T con- 
taming the points a, 6, c, - , I If ar, j5, - , X are positive integers, show that 
the sum of the residues of the function 



fey 



with respect to the poles a, 5, c, , I is a polynomial F(x) of degree 

*v i * a + /3+... + X-l 

satisfying the relations 

J"(a) =/(a), . \ *C- (a) =/<- 
=/(6) f , ^ 



(Make use of the relation ^(x) = /(x) + [/^ <j> (z) dz]/2 m ) 

23* Let/(2) be an analytic function m the interior of a circle C with center 
a On the other hand, let a x , o^, , c^, - - be an infinite sequence of points 
within the circle C, the point a n having the center a for limit as n becomes in- 
finite. Eor every point z within C there exists a development of the form 



126 THE GENERAL CAUCHY THEORY [II, Exs 

where 

F n (z) = (z~a 1 )(z-a 2 ) (*-o) 

[LAURENT, Journal de mathematiques, 5th series, Vol VIII, p 325 ] 
(Make use of the following formula, which is easily verified, 



z x z - ! (z ajte- a 2 ) 

(s-aO (a-fln-i) } . 1 (s-fli) -(s-g*^ 
- - ) (z a,,) ' 



and follow the method used m establishing Taylor's foimula ) 

24 Let z = a + bi be a root of the equation f(z) = X + Ti = of multi- 
plicity w, where the function f(z) is analytic m its neighborhood The point 
x=a, y = b is a multiple point of order n for each of the two cuives JC = 0, 
F = The tangents at this point to each of these cuives form a set of lines 
equally inclined to each other, and each ray of the one bisects the angle between 
the two adjacent rays of the othei 

25 Let /(a) = X + Yi = A Q z + A^- 1 + + A m be a polynomial of the 
mth degree whose coefficients aie numbers of any kind All the asymptotes of 
the two curves X = 0, Y pass through the point -4 1 /mJL and are 
arranged like the tangents in the preceding exercise 

26* Burman's series. Given two functions /(#), F(x) of a variable x, 
Burman's formula gives the development of one of them in powers of the other 
To make the problem more definite, let us take a simple root a of the equation 
F(x) = 0, and let us suppose that the two functions /(x) and F(x) are analytic 
in the neighborhood of the point a In this neighborhood we have 

'"-'is- 

the" function 4>(x) being regular for JG = a if a is a simple root of F(x) = 
Representing F(x) by y, the preceding relation is equivalent to 

x a ?/0 (x) = 0, 
and we are led to develop /(x) in powers of y (Lagrange's formula) 

27*. Kepler's equation The equation z a e sin z 0, where a and e are 
two positive numbers, a < ir, e < 1, has one leal loot lying between and -rr, 
and two roots whose leal parts he between mir and (m + !)TT, wheie m is> any 
positive even mtegei or any negative odd mtegoi If m is positive and odd, 
or negative and even, there aie no roots whose leal parts he between mir and 



[BRIOT ET BOUQUET, TMone desfonctions ellyptiques, 2d ed , p 199 ] 
(Study the curve descubed by the point u = z a esinz when the van- 
able z describes the four sides of the rectangle formed by the stiaight lines 
x = mTT, x =s (m + 1) TT, y = + #, y = JK, where E is very large ) 

28* Foi very large values of m the two roots of the preceding exercise 
whose real parts lie between 2 WTT and (2 m + 1) TT are approximately equal to 
7T/2 * [log (2/e) + log (2 mir + Tr/2)] 

[COURIER, Annafa de VJScole Normale, 2d series, Vol VII, p 73.] 



CHAPTER III 
SINGLE-VALUED ANALYTIC FUNCTIONS 

The first part of this chapter is devoted to the demonstration of 
the general theoiems of Weierstiass* and of Mittag-Leffler on inte- 
gral functions and on single-valued analytic functions with an 
infinite number of singular points TTe shall then make an applica- 
tion of them to elliptic functions 

Since it seemed impossible to develop the theory of elliptic func- 
tions with any degree of completeness in a small number of pages, 
the treatment is limited to a general discussion of the fundamental 
principles, so as to give the reader some idea of the importance of 
these functions. For those who wish to make a thorough study of 
elliptic functions and their applications a simple course in Mathe- 
matical Analysis would never suffice ; they will always be compelled 
to turn to special treatises. 

I. WEIERSTRASS'S PRIMARY FUNCTIONS. MITTAG-LFFLER*S 

THEOREM 

57. Expression of an integral function as a product of primary 
functions. Every polynomial of the rath degree is equal to the prod- 
uct of a constant and m equal or unequal factors of the form x a, 
and this decomposition displays the roots of the polynomial. Euler 
was the first to obtain for sin z an analogous development in an 
infinite product, but the factois of that product, as we shall see fai- 
ther on, are of the second degree in a. Cauchy had noticed that we 
are led in certain cases to ad-join a suitable exponential factor to 
each of the binomial factors such as x a. But Weierstrass was 
the first to treat the question with complete generality by showing 
that every integral function having an infinite number of roots can 
be expressed as the product of an infinite number of factors, each 
of which vanishes for only a single value of the variable. 

* The theorems of Weierstrass which are to be presented here were first published 
in a paper entitled Zur Theone, der eindeutzgen analyttschen Functionen (Berl 
Abhandlungen, 1876, p 11 = Werke, Vol II, p 77). Picard gave a translation of this 
paper in the AnnaUs de PEcote Normale superwure (1879) The collected researches 
of Mittag-Leffler are to be found in a memoir m the Acta mathematics Vol II 

127 



128 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 57 

We alieady know one integral function which does not vanish for 
any value of , that is, e?. The same thing is true of e ff(z > 9 where g(z) 
is a polynomial or an integral transcendental function Conversely, 
every integral function which does not vanish for any value of 2 is 
expressible in that foim In fact, if the integral function G(z) does 
not vanish for any value of 2, every point & = a is an ordinary point 
for G'(z)/G(z) } which is therefore another integral function g^z) : 



) 
Integrating both sides between the limits # , #, we find 



where g(z) is a new integral function of 2, and we have 
G(z) = G(zJtf<*- ff W = 0*C*>-^+ L si: <?(*<,>]. 

The right-hand side is precisely in the desired form. 

If an integral function G (&) has only n roots a v 2 , , a n , distinct 
or not, the function G (2) is evidently of the form 



Let us consider now the case where the equation G(z)=Q has an 
infinite number of roots. Since there can be only a finite number of 
roots whose absolute values are less than or equal to any given num- 
ber R ( 41), if we arrange these roots in such a way that their 
absolute values never diminish as we proceed, each of these roots 
appears in a definite position in the sequence 



where \a n \ ^ |a m+1 |, and where \a n \ becomes infinite with the index n. 
We shall suppose that each root appears in this series as often as is 
required by its degree of multiplicity, and that the root & = is 
omitted from it if <?(0)=: We shall first show how to construct 
an integral function 6^(2) that has as its roots the numbers in the 
sequence (1) and no others. 

The product (1 z/a n }e Q v&, where Q, v (z) denotes a polynomial, is 
an integral function Vhich does not vanish except for & = a n We 
shall take for Q v (z) a polynomial of degree v determined in the fol- 
lowing manner : write the preceding product in the form 



in, 57] PRIMARY FUNCTIONS 129 



and replace Log (1 /a w ) by its expansion in a power series , then 
the development of the exponent will commence with a term of 
degree v + 1, piovided we take 



The integer v is still undetermined We shall show that this number v 
can be chosen as a function of n in such a way that the infinite product 



will be absolutely and uniformly convergent in every circle C of 
radius R about the origin as center, however large R may be The 
radius R having been chosen, let a be a positive number less than 
unity. Let us consider separately, m the product (2), those factors 
corresponding to the roots a n whose absolute values do not exceed 
R/a If there are % roots satisfying this condition, the product of 
these factors 



evidently represents an integral function of Consider now the 
product of the factors beginning with the (# + l)th : 



If % remains in the interior of the circle with the radius R, we 
have [#|^ JR, and since we have \a n \>R/a when n>q, it follows 
that we also have \\ <tf[a w |. A factor of this product can then be 
written, from the manner in which we have taken Q v (z), 



t g 
h_l. 

V < 



if we denote this factor by 1 + M M we have 

l/ay + l 1 /s\y-t-2 

u n==e -T+i(%J M^W/ '1. 

Hence the proof reduces to showing that by a suitable choice of the 
number v the series whose general term is ?7 n = |^ w | is uniformly 
convergent in the circle of radius R (I, 176, 2d ed ). In general, 
if m is any real or complex number, we have 



130 SINGLE-VALUED ANALYTIC FUNCTIONS [ui, 5? 

We liave then, a fortiori, 

1 I Z |" + 1 /14.HI| 2 \4.?1\^\ 2 + A 

7 n ^/+ilM \ l+ + al^r^-j-skl * ;_i ? 
or, noticing that || <a|a n |, when || is less than R, 



But if aj is a real positive number, eF 1 is less than cce*; hence 
we have l 

*~~ v-\~l a n 1 a """ v -J- 1 a tt 1 a 

In order that the series whose general term is U n shall be uni- 
formly convergent in the circle with the radius R, it is sufficient 
that the series whose general term is &/a n v+1 converge uniformly 
in the same circle If theie exists an integer p such that the series 
S|l/# n | y converges, we need only take v =p 1 If there exists no 
integer p that has this property,* it is sufficient to take v = n 1 
For the series whose general term is z/a n \ n is uniformly convergent 
in the circle of radius R, since its terms are smaller than those of 
the senes 2,\R/a n \ n , and the nth root of the general term of this last 
series, or |-R/a n [, approaches zero as n increases mdefinitely.t 

Therefoie we can always choose the integer v so that the infinite 
product -F 2 (#) will be absolutely and uniformly convergent in the 
circle of radius R. Such a product can be replaced by the sum of a 
uniformly convergent series ( 176, 2d ed) whose terms are all 
analytic. Hence the pioduct F^(s) is itself an analytic function 
within this circle (39). Multiplying &%(&) by the product F a (), 
which contains only a finite number of analytic factors, we see that 
the infinite product 



is itself absolutely and uniformly convergent in the interior of the 
circle C with the radius JK, and represents an analytic function within 
this circle. Since the radius R can be chosen arbitrarily, and since 

* For example, let onlog n (w^2) The series whose general term is (log ri)-P 
w divergent, whatever may be the positive number p t for the sum of the first (n - 1) 
terms is greater than (n-l)/0ogn)p, an expression which becomes infinite with n 

f Borel has pointed out that it is sufficient to take for v a number such that v + 1 
shall be greater than logra In fact, the senes S| J2/on|i<>& is convergent, for the 
general term can be written e^e* i<* I -R/i!s*w lo st /!. After a sufficiently large 
value of n, \a n \/K will be greater than &, and the general term less than 1/w 2 . 



Ill, 57] PRIMARY FUNCTION'S 131 

v does not depend on J2, this product is an integral function 6^(2) 
which has as its roots precisely all the various numbers of the 
sequence (1) and no others. 

If the integral function G (z) has also the point z as a root of 
the pth order, the quotient 



is an analytic function which has neither poles nor zeros in the 
whole plane Hence this quotient is an integral function of the form 
e ff& y wheie g(z) is a polynomial or an integial tianscendental func- 
tion, and we have the following expression for the function G (2) 

(4) <?(*) = 

The integral function g(z) can in its tuin be replaced in an infinite 
variety of ways by the sum of a umfoiinly convergent series of 
polynomials 

and the preceding foiinula can be written again 

TT/ 

GW=*II{ -^ 

The factors of this product, each of which vanishes only for one 
value of 2, are called primary functions 

Since the product (4) is absolutely convergent, we can arrange the 
primary functions in an arbitrary order or group them together in 
any way that we please. In this product the polynomials Q v (z) 
depend only on the roots themselves when we have once made a 
choice of the law which determines the number v as a function of n. 
But the exponential factor e g< & cannot be determined if we know 
only the roots of the function G(z) Take, for example, the function 
sin TTZ, which has all the positive and negative integers for simple 
roots In this case the series S f |l/ n | 2 is convergent, hence we can 
take v = 1, and the function 



where the accent placed to the right of n means that we are not to 
give the value zero* to the index n, has the same roots as sin TT#. 

* When this exception is to be made in a formula, we shall call attention to it 
by placing an accent (0 after the symbol of the product or of the sum 



1S2 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 57 

We have then SHUT* = e? w G(), but the reasoning does not tell us 
anything about the factoi e 9(z) We shall show later that this factor 
reduces to the numbei IT 

58. The class of an integral function. Given an infinite sequence 
a v a 2 , , tf n , *, where a n \ becomes infinite with n, we have just 
seen how to construct an infinite number of integial functions that 
have all the terms of that sequence for zeros and no others When 
there exists an integer^; such that the series Sl^l""* is convergent, 
we can take all the polynomials Q v (z) of degree p 1 

Given an integial function of the form 



where P() is a polynomial of degree not higher than p 1, the 
number ^ 1 is said to be the class of that function Thus, the 
function 



is of class zero , the function (sin irz)/ i jr mentioned above is of class 
one. The study of the class of an integial function has given rise in 
recent years to a large number of investigations * 

59. Single-valued analytic functions with a finite number of singular 
points. When a single-valued analytic function F(z) has only a 
finite number of singular points in the whole plane, these singular 
points are necessarily isolated; hence they are poles or isolated 
essentially singular points The point z = oo is itself an ordinary 
point or an isolated singular point ( 52) Conversely, if a single- 
valued analytic function has only isolated singular points in the entire 
plane (including the point at infinity), there can be only a finite 
number of them In fact, the point at infinity is an ordinary point 
for the function or an isolated singular point. In either case we can 
describe a circle C with a radius so large that the function will have 
no other singular point outside this circle than the point at infinity 
itself Within the circle the function can have only a finite number 
of singular points, for if it had an infinite number of them there 
would be at least one limit point ( 41), and this limit point would 
not be an isolated singular point Thus a single-valued analytic 

*See BOREI*, Lemons sur les fonctzons entieres (1900), and the recent work of 
BLTJMBNTHAL, Sur ksfonctions entieres de genre inflni (1910) 



Ill, 59] PRIMARY FUNCTIONS 133 

function which has only jjoles has necessarily only a finite number 
of them, foi a pole is an isolated singulai point 

Every single-valued analytic function which is regular for every 
finite value of z, and for z = oo , is a constant In fact, if the func- 
tion were not a constant, since it is legular for every finite value of 
z, it would be a polynomial or an integral function, and the point at 
infinity would be a pole or an essentially singular point. 

3STow let F(z) be a single-valued analytic function with n distinct 
singular points a v 2 , - ., a n in the finite portion of the plane, and 
let G t [/(z a,)] be the principal part of the development of F(z) 
in the neighborhood of the point a v , then G % is a polynomial or an 
integral transcendental function in l/(z a t ) In either case this 
principal part is regular for every value of z (including & = oo) 
except z == a l Similarly, let P(z) be the principal part of the devel- 
opment of F(z) in the neighborhood of the point at infinity. P(z) 
is zero if the point at infinity is an ordinary point for F(z). The 
difference 



is evidently regular for every value of z including z = GO ; it is there- 
fore a constant C, and we have the equality* 



(5) ^ ) = P(s)+t __ + c , 



which shows that the function F(&) is completely determined, except 
for an additive constant, when the principal part in the neighbor- 
hood of each of the singular points is known. These principal parts, 
as well as the singular points, may be assigned arbitrarily. 

When all the singular points are poles, the principal parts G l are 
polynomials; P(z) is also a polynomial, if it is not zero, and the 
right-hand side of (5) reduces to a rational fraction Since, on the 
other hand, a single-valued analytic function which has only poles 
for its singular points can have only a finite number of them, we 
conclude from this that a single-valued analytic function, all of whose 
singular points are poles, is a rational fraction. 

* We might obtain the same formula "by equating to zero the sum of the residues 
of the function 



where z and z are considered as constants and X as the variable (see 52). 



134 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 60 

60. Single-valued analytic functions with an infinite number of singu- 
lar points. If a single-valued analytic function has an infinite num- 
ber of singular points in a finite region, it must have at least one 
limit point within 01 on the boundaiy of the region For example, 
the function l/sm(l/s) has as poles all the roots of the equation 
sin (l/s)= 0, that is, all the points * = I/&TT, where k is any integer 
whatever The origin is a limit point of these poles Similarly, the 
function 




sin 



has for singular points all the roots of the equation sin (l/#) 
among which are all the points 

2 = __, 

2&'7r-j-aresin( J 

where k and k' are two arbitrary integers All the points l/(2 FTT) 
are limit points, for if, U remaining fixed, k increases indefinitely, 
the preceding expression has l/(2&'7r) for its limit It would be 
easy to construct more and more complicated examples of the same 
kind by increasing the number of sin symbols There also exist, as 
we shall see a little farther on, functions for which every point of a 
certain curve is a singular point. 

It may happen that a single-valued analytic function has only a 
finite number of singular points in every finite portion of the plane, 
although it has an infinite number of them in the entire plane. Then 
outside of any circle C, however great its radius may be, there are 
always an infinite number of singular points, and we shall say that 
the point at infinity is a limit point of these singular points. In the 
following paragraphs we shall examine single-valued analytic func- 
tions with an infinite number of isolated singular points which have 
the point at infinity as their only limit point. 

61. Mittag-Leffler's theorem. If there are only a finite number of 
singular points in every finite portion of the plane, we can, as we 
have already noticed for the zeros of an integral function, arrange 
these singular points in a sequence 

(6) a l9 a 2 , - ., a n , 

in snch a way that we have \a n \ ^ \a n+l \ and that \a n \ becomes infinite 
with n* We may suppose also that all the terms of this sequence 



Ill, 61] PRIMARY FUNCTIONS 135 

are different To each term a l of the sequence (6) let us assign a 
polynomial 01 an mtegial function in l/(s ,)> C?, [!/( a,)], 
taken in an entirely aibitraiy manner. Mittag-Lefflei's theorem may 
be stated thus 

There exists a single-valued analytic function which is regular for 
every finite value ofz that does not occur in the sequence (6), and for 
which the principal part in the neighborhood of the point z = a, is 



We shall prove this by showing that it is possible to assign to 
each function G t [l/( a t )] a polynomial P t () such that the series 



defines an analytic function that has these properties. 

If the point 2 = occurs in the sequence (6), we shall take the 
corresponding polynomial equal to zero. Let us assign a positive 
number e^ to each of the other points a % so that the series Sc, shall be 
convergent, and let us denote by a a positive nuinbei less than unity. 
Let C z be the circle about the origin as center passing through the 
point a lt and C[ the circle concentnc to the preceding with a radius 
equal to a\a t \. Since the function <7 t [l/( a 4 )] is analytic in the 
circle C l? we have for every point within C t 



The power series on the right is uniformly convergent in the circle 
C^ ? hence we can find an integer v so large that we have, in the 
interior of the circle C(, 



Having determined the number v in this manner, we shall take for 
P t (s) the polynomial <r l0 a*iZ a iv z v . 

How let C be a circle of radius R about the point z == as 
center. Let us consider separately the singular points a % in the 
sequence (6) whose absolute values do not exceed R/a. If there 
are q of them, we shall set 



136 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 61 

The remaining infinite series, 



is absolutely and uniformly convergent in the circle <7, since for 
every point in this circle \&\ < R < a|# t | if the index i is greater 
than # From the inequality (7), and from the manner in which we 
have taken the polynomials P % (z), the absolute value of the geneial 
term of the second series is less than ^ when # is within the circle C. 
Hence the function F^z) is an analytic function within this circle, 
and it is clear that if we add *\(z) to it, the sum 



(8) F(z) 

will have the same singular points in the circle C, with the same 
principal parts, as F^z). These singular points are precisely the 
terms of the sequence (6) whose absolute values are less than R, and 
the principal part in the neighborhood of the point a, is GJ[l/(z a t )]. 
Since the radius R may be of any magnitude, it follows that the 
function F(z) satisfies all the conditions of the theorem stated above. 
It is clear that if we add to F(v) a polynomial or any integral 
function whatever G(z), the sum F(z) -f G(z) will have the same 
singular points, with the same principal parts, as the function F(z) 
Conversely, we have thus the general expression for single-valued 
analytic functions having given singular points with corresponding 
given principal parts , for the difference of two such functions, being 
regular for every finite value of , is a polynomial or a transcendental 
integral function Since it is possible to represent the function G(z) 
in turn by the sum of a series of polynomials, the function F(z) + G (z) 
can itself be represented by the sum of a series of which each term 
is obtained by adding a statable polynomial to the principal part 

<?,[!/(* -a,)]- 

If all the principal parts G % are polynomials, the function is 
analytic except for poles in the whole finite region of the plane, and 
conversely. We see, then, that every function analytic except for 
poles can be represented by the sum of a series each of whose terms 
is a rational fraction which becomes infinite only for a single finite 
value of the variable. This representation is analogous to the decom- 
position of a rational fraction into simple elements 

Every function $(z) that is analytic except for poles can also be 
represented by the quotient of two integral functions. For suppose 



Ill, 62] PRIMARY FUNCTIONS 137 

that the poles of $ (z) are the terms of the sequence (6), each being 
counted accoidmg to its degree of multiplicity Let G(z) be an 
mtegial function having these zeros, then the product (z) G(&) 
has no poles It is therefore an integral function 6^(2), and we have 
the equality 



62. Certain special cases. The preceding demonstration of the 
general theorem does not always give the simplest method of con- 
structing a single-valued analytic function satisfying the desired 
conditions Suppose, foi example, it is required to construct a func- 
tion &(z) having as poles of the first order all the points of the 
sequence (6), each residue being equal to unity , we shall suppose 
that 2 = is not a pole. The principal part relative to the pole a, is 
l/(g a t ), and we can write 

_i ^!__ _^ + _l 

2 "** v & 

If we take 



the proof reduces to determining the integer v as a function of the 
index i in such a way that the series 

-foo -. 



shall be absolutely and uniformly convergent in every circle de- 
scribed about the origin as center, neglecting a sufficient number of 
terms at the beginning For this it is sufficient that the series 
S(/a t ) v+1 be itself absolutely and uniformly convergent in the same 
region. If there exists a number^? such that the series S|l/a t |* is 
convergent, we need only take v=p 1. If there exists no such 
integer, we will take as above ( 57) v = i - 1, or v + 1 > log i. The 
number v having been thus chosen, the function 



(9) *(*) 

which is analytic except for poles, has all the points of the sequence 
(6) as poles of the first order with each residue equal to unity. 



188 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 62 

It is easy to deduce from this a new proof of Weierstrass's theorem 
on the decomposition of an integral function into primary functions 
In fact, we can integrate the series (9) term by term along any path 
whatever not passing through any of the poles , for if the path lies 
in a cucle C having its centei at the origin, the series (9) can be 
replaced by a series which is unifoimly convergent in this circle, 
together with the sum of a finite number of functions analytic except 
for poles This results from the demonstration of formula (9) If 
we integrate, taking the point z = Q for the lower limit, we find 



and consequently 



It is easy to verify the fact that the left-hand side of the equation 
(10) is an integral function of &. In the neighborhood of a value a 
of z that does not occur in the sequence (6) the integral f Q *& 
is analytic ; hence the function 



is also analytic and different from zero for z = a. In the neighbor- 
hood of the point ff t we have 



I 

Jo 



where the functions P and Q are analytic It is seen that this inte- 
gral function has the terms of the sequence (6) for its roots, and the 
formula (10) is identical with the formula (3) established above 

The same demonstration would apply also to integral functions hav- 
ing multiple roots If a % is a multiple root of order r, it would suffice 
to suppose that $(#) has the pole & = a % with a residue equal to r 

Let us try again to form a function analytic except for poles of 
the second order at all the points of the sequence (6), the princi- 
pal part in the neighborhood of the point a % being l/(z a^f. We 
shall suppose that z = is an ordinary point, and that the series 
I 8 is convergent, it is clear that the series S|l/<z t | 4 will also 



III, 63] PRIMARY FUNCTIONS 189 

be convergent Limiting the development of !/(# a^ in powers 
of z to its first term, we can write 



and the series 



2a l z-z^ 
A 2 ' 



satisfies all the conditions, provided it is uniformly convergent in 
every circle C descubed about the origin as center, neglecting a 
sufficient number of terms at the beginning ISTow if we take only 
those terms of the series coming from the poles a l for which we have 
\a l \> R/a, R being the radius of the circle C and a a positive num- 
ber less than unity, the absolute value of (1 z/a^)~ z will remain 
less than an upper bound, and the series whose general term is 
2 z/a% 2 /a* is absolutely and uniformly convergent in the circle C, 
by the hypotheses made concerning the poles a z . 

63. Cauchy's method. If F(z) is a function analytic except for poles, 
Mittag-Leffler's theorem enables us to form a series of rational terms 
whose sum F^z) has the same poles and the same principal parts 
as F(z) But it still remains to find the integral function which is 
equal to the difference F(z) F^z) Long before Weierstrass's work, 
Cauchy had deduced from the theory of residues a method by which 
a function analytic except for poles may, under very general condi- 
tions on the function, be decomposed into a sum of an infinite number 
of rational terms It is, moreover, easy to generalize his method 

Let F(z) be a function analytic except for poles and regular in the 
neighborhood of the origin ; and let C 19 C^ , C n? be an infinite 
succession of closed curves surrounding the point & = 0, not pass- 
ing through any of the poles, and such that, beginning with a value 
of n sufficiently large, the distance from the origin to any point what- 
ever of C n remains greater than any given nurnbei. It is clear that 
any pole whatever of F(z) will finally be interior to all the curves 
C n > c *+i> ' ' 9 provided the index n is taken large enough. The 
definite integral 

-J- 



where x is any point within C n different from the poles, is equal 
to F(x) increased by the sum of the residues of F(z)/(& x) with 



140 SINGLE-VALUED ANALYTIC FUNCTIONS [ill, 63 

respect to the different poles of F(z) within C n . Let a L be one of 
these poles. Then the conesponding principal pait G K \l/(z A )] is 
a rational function, and we have in the neighborhood of the point a k 



In the neighboihood of this point we can also write 
1 = 1 = _ 1 _ Z-OL _ (z - atf 

Writing out the product we see that the residue of F(z)/( or) 
with respect to the pole a L is equal to 

__ ^1 ^-m-l ^m . 

We have, then, the relation 



where the symbol S indicates a summation extended to all the poles a k 
within the curve C n On the other hand, we can replace l/(s x) by 



aaid "write the preceding formula m the form 



^ r 

o / 

2 7T^ J ( c n 



(13) 



The integral 



is equal to F(0) increased by the sum of the residues of F(z)/z with 
respect to the poles of F(z) within C n . More generally, the definite 
integral 

1 C 

27r Vcc tt 
is equal to 



plus the sum of the residues of tr r F(&) with respect to the poles of 
F(z) within C n . If we represent by 4 r -*> the residue of z~ r F(z) 



m, es] 



PRIMARY FUNCTIONS 



141 



relative to the pole a L , -we can wiite the equation (13) m the form 



(14) 



*(<>) + ? 



_L. r m/* 

2m J(c^^ x ^ 



In order to obtain an upper bound for the last term, let us -write 
it in the foim 



= ^i! C 

2 *Ac n 



Let us suppose that along C n the absolute value of sr p F(z) remains 
less than M, and that the absolute value of z is greater than 8 Since 
the number n is to become infinite, we may suppose that we have 
already taken it so laige that 8 may be taken greater than |#|; hence 
along C n we shall have 

1 ^ 1 



If S n IB the length of the curve C n) we have then 

M* +1 



We shall have proved that this term R n approaches zero as n becomes 
infinite if we can find a sequence of closed curves C v C z> > , C n> - 
and a positive integer^? satisfying the following conditions: 

1) The absolute value of z~ p l r (ii) remains less than a fixed num- 
ber M along each of these curves. 

2) The ratio S n /8 of the length of the curve C n to the minimum 
distance 8 of the origin to a point of C n remains less than an upper 
bound Z as n becomes infinite 

If these conditions are satisfied, \R n \ is less than a fixed number 
divided by a number 8 \x \ which becomes infinite with n. The term 
R n therefore approaches zero, and we have in the limit 



(15) 



Thus we have found a development of the function F(x) as a sum 
of an infinite series of rational terms The order in which they occur 



142 



SINGLE- VALUED ANALYTIC FUNCTIONS [III, 63 



in the seiies is determined by tlie arrangement of the cuives 
CO- C in their sequence If the series obtained is abso- 
lutely 2 convergent, we can mite the teims in an arbitiary order 

Note. If the point = were a pole for -F(*0 with the pimeipal 
part <?(!/*), it would suffice to apply the preceding method to the 
function F(z) - G (I/*). 

64. Expansion of ctn* and of sin* Let us apply this method to 
the function F() =ctn *!/*, which has only poles of the first oider 
at the points * = ITT, where k is any integer diif erent from zero, the 
residue at each pole being equal to unity We shall take for the 
curve C n a square, such as BCB'C', having the ougin for center and 
having sides of length 2nir + ir parallel to the axes , none of the 
poles are on this boundary, and the ratio of the length S n to the 
minimum distance S from the origin to a point of the boundary 
is constant and equal to 8. The squaie of the absolute value of 

ctn (x + yi) is equal to 



O 



nir 



On the sides BC and B'C 1 we have 
cos 2 a? = 1, and the absolute value 
is less than 1. On the sides BB' and 
CO 1 the square of this absolute value 
is less than 



FIG. 23 



We must replace 2 y in this formula by (2 w + 1) TT, and the ex- 
pression thus obtained approaches unity when n becomes infinite. 
Since the absolute value of l/ along C n approaches zero when n 
becomes infinite, it follows that the absolute value of the function 
ctn 1/3 on the boundary C n remains less than a fixed number M, 
whatever n may be Hence we can apply to this function the for- 
mula (15), taking^ = 0. We have here 



~ ,K T 
F(Q) = hm 

^ / * 



x cos x sin x 
x since 



\ A 

) = 0, 

/ 



and 4 ? which represents the residue of (ctn 3 !/)/ 
kir, is eqiial to I/^TT. We have, then, 



(16) 



1 n / -I -J > 

cta-ilimV'( V~ + r~ 

X noori w * 7r ;7r ' 



Ill, 64] PRIMARY FUNCTIONS 143 

where the value k = is excluded from the summation The infinite 
series obtained by letting n become infinite is absolutely convergent, 
for the geneial teirn can be written in the form 



x LIT ATT A/TT (A-7T a;) 

and the absolute value of the factor a*/(l x/kir) remains less than 
a certain uppei bound, provided x is not a multiple of TT We have, 
then, precisely 

(IT) 

Integrating the two members of this relation along a path start- 
ing from the origin and not passing through any of the poles, we find 



from which we derive 
(18) am* 



The factor &&> is here equal to unity If in the series (17) we combine the 
two terms which come from opposite values of k, we obtain the formula 



Combining the two factors of the product (18) which correspond to opposite 
values of fc, we have the new formula* 



or, substituting irx for 



Note 1 The last f ormulsa show plainly the periodicity of sm x, which does 
not appear from the power series development. We see, in fact, that (sin wa;)/7r 
is the limit as n becomes infinite of the polynomial 



* This decomposition of sin x into an infinite product is due to Enler, who obtained 
at m an elementary manner (Introductto ^n Analy&in infimtorwn) 



SINGLE-VALUED ANALYTIC FUNCTIONS [III, 64 
Replacing x by x + 1, this formula may be written in the form 



whence, letting n become infinite, we find sm (irx + TT) = sinine, or 

sin (2 + TT) = sins;, 

and therefore sin (z + STT) = sing 

Note 2. In this particular example it is easy to justify the necessity of associ- 
ating with each binomial factor of the form 1 or/a*, a suitable exponential factoi 
if we wish to obtain an absolutely convergent product For defimteness let us 
suppose x real and positive The series Zx/n being divergent, the product 



becomes infinite with m, while the product 



K) 

(-3 



approaches zero as n becomes infinite (I, 177, 2d ed ) If we take m = n, the 
product P m Qm has (SHITTX)/*- for its limit , but if we make m and n become 
infinite independently of each other, the limit of this product is completely in- 
determinate This is easily verified by means of Weierstrass's primary functions, 
whatever may be the value of x Let us note first that the two infinite products 



are both absolutely convergent, and their product Fi(x)F 9 (x) is equal to (sixnnc)/7r. 
With these facts m mind, let us write the product P m Q n in the form 



When the two numbers m and n become infinite, the product of all the fac- 
tors on the right-hand side, omitting the last, has F T (x) F z (x) = (smirx)/ir for its 
limit. As for the last factor, we have seen that the expression 



has for its limit log , where w denotes the limit of the quotient m/n (I, 161) 
The product P m Q has, therefore, 



for its limit. Hence we see the manner m which that limit depends upon the 
law according to which the two numbers m and n become infinite. 

Note 3. We can make exactly analogous observations on the expansion of ctn x 
We shall show only how the periodicity of this function can be deduced from the 
series (17). Let us notice first of all that the series whose general term is 



Ill, 65] 



ELLIPTIC FUNCTIONS 



145 



where the index fc takes on all the integral values from oo to 4- *>, excepting 
k = 0, k = 1, is absolutely convergent , and its sum is 2/7r, as is &een on mat- 
ing k vary first f lorn 2 to + oo, then from 1 to oo "We can therefore write 
the development of ctnx in the form 



x - 



x J 



where the values fc = 0, k I are excluded from the summation This results 
from subtracting from each term of the series (17) the corresponding term of 
the convergent series formed by the preceding series togethei with the additional 
term 2/V Substituting x + IT f or sc, we find 



or, again, 



where fc 1 takes on all integral values except 
identical with etna. 



The light-hand side is 



II DOUBLY PERIODIC FUNCTIONS ELLIPTIC FUNCTIONS 

65. Periodic functions. Expansion in senes. A single-valued analytic 
function /() is said to be periodic if there exists a real or complex 
number o> such that we have, whatever may be z,f(& + w)=j*(*)> 
this number o> is called 
a period. Let us mark 
in the plane the point 
representing <o, and let 
us lay off on the unlim- 
ited straight line pass- 
ing through the origin 
and the point o> a length _ 
equal to | | any number 
of times in both direc- 
tions. We obtain thus 
the points o>, 2 o>, 3 o>, 
- t w<o, - and the 
points CD, 2 o>, - , 
wo), .-. Through 
these different points 
and through the origin let us draw parallels to any direction differ- 
ent from Ov, the plane is thus divided into an infinite number of 
cross strips of equal breadth (Fig. 24) 







FIG 24 



146 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 65 

If through any point z we draw a parallel to the direction <9co, we 
shall obtain all the points of that straight line by allowing the leal 
parameter X in the expiession z 4- Xo> to vary from QQ to + o In 
particular, if the point z describes the fiist strip AA } BE^ the corre- 
sponding point z -f (o will describe the contiguous stup BB'CC 1 , the 
point 3 -f- 2 w will describe the thud strip, and so on in this manner. 
All the values of the f unction /(#) in the first strip will be duplicated 
at the corresponding points in each of the other strips 

Let LV and MM 1 be two unlimited straight lines parallel to the 
direction Oo Let us put u = e 2tirs ' u , and let us examine the region 
of the ^-plane described by the vanable u when the point z remains 
in the unlimited cross strip contained between the two parallels LL* 
and J/A/' If a -f /3ns a point of LL\ we shall obtain all the other 
points of that straight line by putting z = a + f$i + X<o and making 
X vary from oo to -f oo Thus, we have 



hence, as X varies from oo to -f co , u describes a circle C l having the 
origin for center Similaily, we see that as & describes the straight line 
J/fl/', u remains on a circle C 2 concentric with the first , as the point 
describes the unlimited strip contained between the two straight 
lines LV, MM*, the point u describes the ring-shaped region contained 
between the two circles C 19 C r But while to any value of # there 
corresponds only one value of u 9 to a value of u there correspond an 
infinite number of values of & which form an arithmetic progression, 
with the common difference o>, extending forever in both directions 
A periodic function /(), with the period CD, that is analytic in the 
infinite cross strip between the two straight lines LL 1 , MM*, is equal 
to a function <() of the new variable u which is analytic in the 
ring-shaped region between the two circles C^ and C 2 . For although 
to a value of u there correspond an infinite number of values of #, 
all these values of s give the same value to f(z) on account of its 
periodicity. Moreover, if U Q is a particular value of u, and s? any 
corresponding value of , that determination of 3 which approaches 
ZQ as u approaches U Q is an analytic function of u in the neighbor- 
hood of ; hence the same thing is true of <(w). We can therefore 
apply Laurent's theorem to this function < (w) In the ring-shaped 
region contained between the two circles C v C 2 this function is 
equal to the sum of. a series of the following form : 



HI, 66] ELLIPTIC FUNCTIONS 147 

Eeturning to the variable z, we conclude from tins that in the in- 
terior of the cross strip consideied above the periodic function f(z) 
is equal to the sum of the series 



If the function/^) is analytic in the whole plane, we can suppose 
that the two straight lines LL\ J/J/', which bound the stiip, recede 
indefinitely in opposite directions Eveiy periodic integral function 
^s therefore developable vn, a series of positive and negative poicers of 
convergent for every finite value of z 



66 Impossibility of a single-valued analytic function with three periods. By a 
famous theorem due to Jacobi, a single-valued analytic function cannot have 
more than two independent periods To prove this we shall show that a single- 
valued analytic function cannot have three independent periods * Let us first 
prove the following lemma 

Let a, 6, c be any three real or complex quantities, and m, n, p three arbi- 
trary integers, positive or negative, of which one at least is different from zero. 
If we give to the integers m, w, p all systems of possible values, except 

m = n = p = 0, 

the lower limit of \ ma + nb -{- pc \ is equal to zero 

Consider the set (E) of points of the plane which represent quantities of the 
form ma + rib -f pc If two points corresponding to two different systems of 
integers coincide, we have, for example, 

ma + rib + pc = m t a + nfi + p^ 
and therefore 

(m - m x ) a + (n - njb + (p-pjc = 0, 

where at least one of the numbers m m x , n 74, p p x is not zero In this 
case the truth of the lemma is evident. If all the points of the set (E) are dis- 
tinct, let 2 8 be the lower limit of [ma + nb + pc \ , this number 2 S is also the 
lower limit of the distance between any two points whatever of the set (E) In 
fact, the distance between the two points ma -f nb -f pc and m 1 a + n^ 4- p^ is 
equal to | (m m x ) a + (n n t ) b + (p pj c |. We are going to show that we 
are led to an absurd conclusion by supposing $ > 0. 

Let N be a positive integer , let us give to each of the integers m, n, p one 
of the values of the sequence N, (N 1), , 0, - - , N 1, N, and let 
us combine these values of m, n, p, m all possible manners. We obtain thus 
(2 N+ I) 8 points of the set (^), and these points are all distinct by hypothesis 
Let us suppose |a|^|6|^|c|; then the distance from the origin to any one 
of the points of (E) just selected is at most equal to 3 N\ a \ These points there- 
fore lie in the interior of a circle O of radius BN\a\ about the origin as center 
or on the circle itself. If from each of these points as center we describe a 

* Three periods a, &, c are said to be dependent if there exist three integers m, n, p 
(not all zero) for which ma + rib +pc~Q TRANS 



148 SINGLE-VALUED ANALYTIC FUNCTIONS [ill, 66 

circle of ladms 5, all these circles will be mtenor to a cncle O x of radius equal 
to 3JY|a| + 5 about the origin as centei, and no two of them will overlap, since 
the distance between the centers of two of them cannot be smaller than 2 S The 
sum of the areas of all these small cucles is therefore less than the area of the 
cncle <?-,, and we have 

or 



(3JZV+1)*-! 

The right-hand side approaches zero as JST becomes infinite , hence this in- 
equality cannot be satisfied foi all values of N by any positive number 5 
Consequently the lower limit of \ma + rib + pc \ cannot be a positive number , 
hence that lower limit is zeio, and the truth of the lemma is established 

We see, then, that when no systems of integers m, n, p (except m = n = p = 0) 
exist such that ma -f nb + pc = 0, we can always find integral values for these 
numbers such that \ma + rib + pc\ will be less than an arbitrary positive num- 
ber e In this case a single-valued analytic f unction f(z) cannot have the three 
independent periods a, 6, c For, let z be an ordinary point for/(z), and let 
us describe a circle o radius e about the point Z Q as centei, where e is so 
small that the equation f(z) =/( ) has no other root than z = z inside of this 
circle ( 40) If a, &, c are the periods of /(z), it is cleai that ma + nb 4- pc is 
also a period for all values of the integers m, n, p , hence we have 

/(* + met + rib + pc) =/(z ) 

If we choose m, n, p in such a manner that | ma -f rib + pc \ is less than , the 
equation f(z) =/(z ) would have a root z 1 different from 2 , where \z z \<e, 
which is impossible. 

When there exists between a, 6, c a relation of the form 

(20) 7na + rib+pc = Q, 

without all the numbers m, n 7 p being zero, a single-valued analytic function 
f(z) may have the periods a, 6, c, but these periods reduce to two periods or to 
a single period. We may suppose that the three integers have no common divisor 
other than unity Let JD be the greatest common divisor of the two numbers 
m, n , m = Dm', % = Dn' Since the two numbers m', n' are prime to each other, 
we can find two other integers m", n" such that m'n" m'V = 1 Let us put 

m'a + rib = a', m"a + n"b = &', 

then we shall have, conversely, a = n"af w/o', b = m'6 7 m"af If a and 6 are 
periods of /(), a" and b' are also, and conversely Hence we can replace the 
system of two periods a and b by the system of two periods a' and V The re- 
lation (20) becomes Daf -f pc = , D and p being prime to each other, let us 
take two other integers & and p' such that Dp" ~ D'p = 1, and let us put 
I/of -t- #'c = c' We obtain from the preceding relations of = pc', c = DC', 
whence it Is obvious that the three periods a, 6, c are linear combinations of the 
two penods 6 7 and c' 



As a corollary of the preceding lemma we see that if a and ft are two 
real quantities and m, n two arbitrary integers (of which at least one is not zero), 
the lower limit of | ma + np | is equal to zero For if we put o = a, & = , c i, 



Ill, 67] 



ELLIPTIC FUNCTIONS 



149 



the absolute value of ma -f- up + pi can be less than a number e < 1 only if we 
have p = 0, | ma -f n| <e Piom this it follows that a single-valued analytic 
function f(z) cannot have two leal independent periods a. and $ If the quotient 
/5/a is iriational, it is possible to find tvv o numbers m and n such that [ma -f n/5 1 
is less than , and it will be possible to carry thiough the reasoning just as 
before If the quotient $/a is rational and equal to the irreducible fraction m/n, 
let us choose two integeis m! and n' such that mn' m'n = 1, and let us put 
m'a n'p = y The number 7 is also a period, and from the two relations 
manp^ 0, m'an'p = 7 we derive or := 117, /S = 9717, so that a and /3 
are multiples of the single period 7 More generally, a single-valued analytic 
function /() cannot have two independent periods a and 6 whose ratio is real, 
for the function f(az) would have the two real penods 1 and &/* 

67. Doubly periodic functions. A doubly periodic function is a 
single-valued analytic function having two periods whose ratio is 
not real. To conform to "Weierstrass's notation, we shall indicate the 
independent vanable by u 9 the two periods by 2 and 2 / 3 and we 
shall suppose that the coefficient of i in w'/o) is positive. Let us 
mark in the plane the points 2 CD, 4 >, 6 <>, and the points 2 <o f , 
4 w', 6 o>', ... Through the points 2 mo let us draw parallels to the 







FIG 25 

direction 0o> r , and through the points 2mV parallels to the direc- 
tion 0(D. The plane is divided in this manner into a net of 
congruent parallelograms (Eig. 25) Let f(u) be a single-valued 
analytic function with the two periods 2o>, 2 a/; from the two 
relations f(u + 2 o>) =/(V), f(u + 2& l )t=sf(u) we deduce at once 

* It is now easy to prove that there exists for any periodic single-valued f unction 
at least one pair of periods in terms of which any other period can be expressed as an 
integral linear combination , such a pair is called a primitive pair of per tods TRANS. 



150 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 67 



f(u + 2 m<*> -f 2 ?H V) = /(ze), so that 2 ?wo> 4- 2 ?>tV is also a period 
for all yalues of the integeis m and m' We shall repiesent this 
general penod by 2w 

The points that lepresent the various periods are precisely the 
vertices of the preceding net of paiallelograins When the point u 
describes the parallelogram OABC whose vertices are 0, 2 <o, 2 w 4- 2 <*>', 
2 o) r , the point ^t, + 2w describes the parallelogiani whose vertices 
are the points 2 w, 2 w + 2 o>, 2 ?0 + 2 o> -h 2 o>', 2w + 2 o>' ? and the 
function /(w) takes on the same value at any pair of coirespondmg 
points of the two parallelograms. Every parallelogram whose ver- 
tices are four points of the type , w + 2 , + 2 ', w + 2 w + 2 co' 
is called a parallelogram of periods , in general we consider the 
parallelogram OABC, but we could substitute any point in the plane 
for the oiigm. The period 2 w + 2 <o f will be designated for brevity 
by 2 <D"; the center of the paiallelogram OABC is the point o>", while 
the points <o and CD' are the middle points of the sides OA and OC 

Every integral doully periodic function is a constant In fact, let 
f(u) be a doubly periodic function , if it is integral, it is analytic in 
the parallelogram OABC, and the absolute value of f(ii) remains 
always less than a fixed number M in this parallelogram But on 
account of the double periodicity the value of /(w) at any point of the 
plane is equal to the value of f(u) at some point of the parallelogram 
OABC. Hence the absolute value of f(u) remains less than a fixed 
number M It follows by Liouville's theorem that f(u) is a constant. 

68. Elliptic functions. General properties. It follows from the pre- 
ceding theorem that a doubly periodic function has singular points 
in the finite portion of the plane, unless it reduces to a constant. 
The term elliptic function is applied to functions which are doubly 
periodic and analytic except for poles In any parallelogram of 
penods an elliptic function has a certain number of poles , the num- 
ber of these poles is called the order of the function, each being 
counted according to its degree of multiplicity *. It should be noticed 
that if an elliptic function f(u) has a pole U Q on the side OC, the 
point U Q 4- 2 w, situated on the opposite side AB, is also a pole , but 
we should count only one of these poles in evaluating the number 
of poles contained m OABC. Similarly, if the origin is a pole, all the 

* It is to be understood that the parallelogram is so chosen that the order is as 
small as possible Otherwise, the number of poles in a parallelogram could be taken to 
be any multiple of this least number, since a multiple of a period is a period TRANS 
(See also the footnote, p 149 ) 



Ill, 



ELLIPTIC FUNCTIONS 



151 



veitices of the net are also poles off(u\ but we should count only 
one of them in each parallelogram If, foi example, we move that 
vertex of the net which lies at the origin to a suitable point as near 
as we please to the origin, the given function /(K) no longei has 
any poles on the boundaiy of the parallelogiam When we have occa- 
sion to integrate an elliptic function /() along the boundary of the 
parallelogram of periods, we shall alwa} s suppose, if it is necessary, 
that the parallelogram has been displaced in such a way that f(u) 
has no longer any poles on its boundaiy The application of the 
general theorems of the theoiy of analytic functions leads quite 
easily to the fundamental piopositions 

1) The sum of the residues of an elliptic function with respect 
to the poles situated in a parallelogram of periods is zero 

Let us suppose for definiteness that f(ii) has no poles on the 
boundary OABCO The sum of the residues with respect to the poles 
situated within the boundary is equal to 



the integral being taken along OABCO But this integral is zero, for 
the sum of the integrals taken along two opposite sides of the paial- 
lelogram is zero Thus we have 



c 

J(OA) 



and if we substitute u -f- 2 o>' for u in the last integial, we have 

r f(u)du= c f(u+2u r )du=* f/(w)<fo=- c /oo 

*/() *A ^2toi J(OA) 

Similarly, the sum of the integrals along AB and 
along CO is zero In fact, this piopeity is almost 
self-evident from the figure (Fig 26). For let us 
consider two corresponding elements of the two inte- 
grals along OA and along BC* At the points m and 
m 1 the values of f(u) are the same, while the values 
of du have opposite signs 

The preceding theorem proves that an elliptic func- 
tion f(u) cannot have only a single pole of the first 
order in a parallelogram of periods. An elliptic function is at least 
of the second order, 




FIG. 



152 SINGLE-VALUED ANALYTIC FUNCTIONS [HI, 68 

2) The number of zeros of an elliptic function m a parallelogram 
of periods is equal to the order of that function (each of the zeros 
being counted according to its degree of multiplicity) 

Let/(?/) be an elliptic function, the quotient /(w)//() == </>(ii) is 
also an elliptic function, and the sum of the residues of <f> (it) in a par- 
allelogram is equal to the number of zeros of f(if) diminished by the 
numbei of the poles ( 48) Applying the pieceding theoiem to the 
function <(?/), we see the truth of the proposition ]ust stated In gen- 
eial, the numbei of roots of the equation f(it) = C in a paiallelogram 
of periods is equal to the oidei of the function, for the function 
/(?/) C has the same poles as /()> whatever may be the constant 0. 

3) The difference between the sum of the zeros and the sum of the 
poles of an elliptic function in a parallelogram of periods is equal to 
a period. 

Consider the integral 



1 r 

o I 

2-iriJ 



, 

u ^7~i du 
/(M) 

along the boundary of the parallelogram OAEC This integral is 
equal, as we have already seen ( 48), to the sum of the zeios of /() 
within the boundaiy, diminished by the sum of the poles of /(%) 
within the same boundary. Let us evaluate the sum of the integrals 
lesulting from the two opposite sides OA and EC 



r 

Jo 



o 

If we substitute u + 2 <a' for u in the last integral, this sum is equal to 



or, on account of the periodicity of f(u), to 
The integral 



f 

Jo 



is equal to the variation of Log[/(w)] when u describes the side OA\ 
but since /(w) returns to its initial value, the variation of Log [/(%)] 
is equal to 2 m 2 iri 9 where m 2 is an integer The sum of the inte- 
grals along the opposite sides OA and EC is therefore equal to 



Ill, 68] ELLIPTIC FUNCTIONS 153 

(4 M 2 7rio> r )/2 iri = 2???X Similarly, the sum of the integrals along 
AB and along CO is of the form 2 m^ The diffeience considered 
above is theiefoie equal to 2 m^ + 2 w 2 co' , that is, to a period 

By a smiilai argument it can be shown that the proposition is 
also applicable to the loots of the equation /(w) = C, contained in a 
parallelogram of penods, foi any value of the constant C. 

4) Between any two elliptic functions with the same periods there 
exists an algebraic relation 

Let /(?/), / x (w) be two elliptic functions with the same periods 
2o>, 2co'. In a parallelogram of periods let us take the points a l9 
a^ , a m which aie poles for either of the two functions f(u), 
/ x (w) or foi both of them, let^t, be the higher oider of multi- 
plicity of the point a t with lespect to the two functions, and let 
A& x + Aj + + A& m = N. ^ow let F(v, y) be a polynomial of degree n 
with constant coefficients If we replace x and y by f(it) and f^u*), 
respectively, in this polynomial, theie will lesult a new elliptic func- 
tion $ (it) which can have no othei poles than the points a l9 # , - , a m 
and those which are deducible from them by the addition of a period. 
In order that this function <>(?*) may reduce to a constant, it is 
necessaiy and sufficient that the principal paits disappeai in the 
neighborhood of each of the points a v a 3 , - -, a m . Now the point a l 
is a pole for < (u) of an order at most equal to n^ Wntmg the con- 
ditions that all the principal parts shall be zero, we shall have then, 
in all, at most 

n Oi + A*a + ---- M) = Nn 

linear homogeneous equations between the coefficients of the poly- 
nomial F(x, y) in which the constant teim does not appear There 
are n(n + 3)/2 of these coefficients, if we choose n so large that 
n(n + 3)> 2Nn, or n + 3 > 2 X, we obtain a system of linear 
homogeneous equations in which the number of unknowns is greater 
than that of the equations Such equations have always a system of 
solutions not all zero. If F(x 9 y) is a polynomial determined by 
these equations, the elliptic functions f(u\ f^u) satisfy the algebraic 
relation 



where C denotes a constant 

Notes. Before leaving these general theorems, let us make some 
further observations which we shall need later 

A single-valued analytic function /(?) is said to be even if we 
have /( ) =/(w) , it is said to be odd if we have /( w) =/(). 



154 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 68 

The derivative of an even function is an odd function, and the 
derivative of an odd function is an even function In geneial, the 
derivatives of even older of an even function are themselves even 
functions, and the derivatives of odd older are odd functions. On 
the contrary, the deiivatives of even order of an odd function are 
odd functions, and the deiivatives of odd order aie even functions 

Let/(w) he an odd elliptic function, if w is a half-peiiod, we 
must have at the same time /<V)=-/(- w) and/(w)=/(- w), 
since % = iff + 2 w K 1S necessaiy, then, that f(w) shall be zero 
or infinite, that is, that w must be a zero 01 a pole for f(u) The order 
of multiplicity of the zeio 01 of the pole is necessarily odd, if w 
weie a zeio of even order 2 for/(), the derivative /< 2n) (>), which 
is odd, would be analytic and different from zero for u =? w If w 
were a pole of even order for/(w), it would be a zero of even order 
foi 1/f (u). Hence we may say that every half-period is a zero or a, 
pole of an odd order for any odd elliptic function. 

If an even elliptic function /(u) has a half-period w for a pole or 
for a zero, the order of multiplicity of the pole or of the zero is an 
even number If, for example, w were a zero of odd order 2 n + 1, it 
would be a zero of even order for the derivative /'(*), which is an 
odd function The proof is exactly similar for poles Since twice a 
period is also a period, all that we have just said about half-periods 
applies also to the periods themselves. 

69. The function p(w). We have already seen that every elliptic 
function has at least two simple poles, or one pole of the second order, 
in a parallelogram of periods In Jacobi's notation we take func- 
tions having two simple poles for our elements ; in Weierstrass's 
notation, on the contrary, we take for our element an elliptic func- 
tion, having a single pole of the second order in a paiallelogram 
Since the residue must be zero, the principal part in the neighbor- 
hood of the pole a must be of the form. A/(u df. In older to 
make the problem completely definite, it suffices to take A=l and 
to suppose that the poles of the function are the origin u = and 
all the vertices of the network 2w = 2mo> + 2mV. We are thus 
led first to solve the following problem : 

To form an elliptic function having as poles of the second order all 
the points 2 w = 2 m<o + 2 m'w', where m and m! are any two integers 
whatever, and having no other poles, so that the principal part in the 
neighborhood of the point 2 w shall be l/(w 



Ill, 69] ELLIPTIC FUNCTIONS 155 

Before applying to this problem the geneial method of 62, we shall 
first prove that the double senes 



where m and m 1 take on all the integral values fiom oo to 4- oo 
(the combination m = m' = being excepted), is convergent, provided 
that the exponent p is a positive number greater than 2. Consider the 
triangle having the thiee points u = 0, u = wo>, u = ma 4- mV for 
its vertices , the lengths of the three sides of the triangle are respec- 
tively |mo>|, |mV|, \m<o + wV| We have, then, the relation 



where is the angle between the two directions 0&, 0o> f (0 < B < TT) 
For bievity let ||= a, |o>'| = 5, and let us suppose a ^. The pie- 
ceding relation can then be written in the form 

|mo> 4 mV j 3 = ra 2 a 2 -f w' 2 Z 2 2 mm'ab cos <>, 

where the angle is equal to if ^ Tr/2, and to TT ^ if > Tr/2 
The angle cannot be zero, since the three points 0, CD, <o f are not 111 
a straight line, and we have ^ cos < 1. "We have, then, also 

|m<o + m r co r | 2 = (1 cos <>) (w 2 a 2 + m 12 ^ 2 ) 4- cos (ma m f ^) 2 , 
and consequently 
| mo 4- m'<o f | 2 ^ (1 cos ) (m 2 2 4 w' 2 ^ 2 ) S (1 cos ) a 2 (w 2 -f m 12 ). 

From this it follows that the terms of the senes (21) are respectively 
less than or equal to those of the senes S'l/(#t 2 -f m 18 )'* 72 multiplied 
by a constant factor, and we know that the last series is convergent 
if the exponent p/2 is greater than unity (I, 172) Hence the 
senes (21) is convergent if we put /<& = 3 01 p> = 4. According to a 
result derived in 62, the series 



represents a function that is analytic except for poles, and that has 
the same poles, with the same principal parts, as the elliptic function 
sought. We shall show that this function <f> (u) has precisely the two 
periods 2 o> and 2 a/ Consider first the series 



where 2 w = 2 m<# 4- 2 m f <D f 3 the summation being extended to all the 
-integral xstLiies of m and m', except the combinations m = m f = 



156 SINGLE-VALUED ANALYTIC FUNCTIONS [in, 69 

and m = 1, m T = 0. This series is absolutely convergent, for it 
lesults from the series $(ii) when we substitute 2<o foi u and 
omit two teims. It is easily seen that the sum of this series is zero 
by considering it as a double series and evaluating separately each 
of the rows of the rectangular double array Subtracting this series 
from <(V), we can then wiite 



i 

<o) 2 J' 



the combinations (w=w r = 0), (m = 1, m' = 0) being always 
excluded from the summation Let us now change u to u 2 <o , 
then we have 



the combination m = 1, ra f = being the only one excluded fiom 
the summation. But the right-hand side of this equality is identical 
with (w). This function has therefoie the period 2 o>, and in like 
manner we can prove that it has the period 2 o>' This is the func- 
tion which Weierstrass represents by the notation p(w), and which 
is thus defined by the equation 



(22) P() = + 

If we put u = in the difference p (u) 1/w 2 , all the terms of the 
double sum are zero, and that difference is itself zero The function 
p(^) possesses, then, the following properties 

1) It is doubly periodic and has for poles all the points 2 w and 
only those. 

2) The principal part in the neighborhood of the origin is 1/M 2 

3) The difference p(w) 1/w 2 is zero for u = 0. 

These properties characterize the function p (u) In fact, any analy- 
tic function f(u) possessing the first two properties differs from p (u) 
only by a constant, since the difference is a doubly periodic func- 
tion without any poles. If we have also f(u) 1/V = for u = 0, 
f(u) p(u) is also zero for u = ? we have, therefore, /(^) = p(w). 

The function p( w) evidently possesses these three properties; 
we have, then, p( u)= p(w), and the function p(-w) is even, which 
is also easily seen from the formula (22) 

Let us consider the period of p (u) whose absolute value is smallest, 
and let 8 be its absolute value. Within the circle C s with the radius 
B 3 described about the origin as center, the difference p(u) 1/u? is 



Ill, 69] ELLIPTIC FUNCTIONS 157 

analytic and can be developed in positive powers of u The general 
term of the series (22), developed in poweis of u, gives 

T + - 5 



4 w? 2 (2 zr) 3 ^ (2 w) 4 ^ ^ (2 w 
and it is easy to prove that the function. 

5 u 

16|wf u 

M 

dominates this series in a circle of radius 3/2, and, a fortiori, the 
expression obtained from it by replacing 1 u/\w\ by 1 2-w/S 
dominates the series Since the senes S'l/jwj 8 is convergent, we 
have the right to add the resulting senes term by term (9). The 
coefficients of the odd powers of u are zero, for the terms resulting 
from periods symmetrical with respect to the origin cancel, and we 
can write the development of p (u) in the form 

(23) p(u)=^ + %u* + c z i<* + - + c x u**-* + --, 
where 




(24) 



Whereas the formula (22) is applicable to the whole plane, the new 
development (23) is valid only in the interior of the circle C B hav- 
ing its center at the origin and passing through the nearest vertex 
of the periodic network. 

The derivative p'(w) is itself an elliptic function having all the 
points 2w foi poles of the third ordei. It is represented in the 
whole plane by the series 

(25) pW= _2_ 



In general, the nth derivative p (n) (?/) is an elliptic function having 
all the points 2 w for poles of order n + 2, and it is represented by 
the series 



(26) 

We leave to the reader the verification of the correctness of these 
developments, which does not present any difficulty in view of the 
properties established above ( 39 and 61) 



158 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 70 

70. The algebraic relation between p(i/) and p'(i/). By the general 
theorem of 68 theie exists an algebraic relation between p(w) and 
p'(w). It is easily obtained as follows In the neighborhood of the 
origin we have, from the foiinula (23), 



where the terms of the senes not written are zero for u = The 

difference ?*() - 4 p*( w ) has tn ^ refore the 011 S m as a P ole of tlie 
second order, and in the neighborhood of this point we have 



where the terms not written are zeio for u = 

Hence the elliptic function 20 cjp(u) 28 c z has the same poles, 
with the same principal parts, as the elliptic function p' 2 - 4p 8 , and 
their difference is zero when u = These two elliptic functions are 
therefore identical, and we have the desired idation, which we shall 
write in the form 



(27) [p'(f O] 1 = 4 pF() - ff j> (u) - 

where 



The relation (27) is fundamental in the theory of elliptic func- 
tions ; the quantities g 2 and g z are called the invariants 

AH the coefficients o x of the development (23) are polynomials in 
terms of the invariants g 2 and g 9 In fact, taking the derivative of 
the relation (27) and dividing the result by 2 P'(M), we derive the 
formula 



(28) p 

On the other hand, we hare in the neighborhood of the origin 

"() = 1 + 2e a + 12e^+ . . -f-(2X- 2)(2X - 8)^*-*+ 



m,7l] ELLIPTIC FUNCTIONS 159 

Replacing p(w) and p"(w) by their developments in the relation 
(28), and remembering that (28) is satisfied identically, we obtain 
the recurrent relation 



Cti *? Q /\ Q\T 

V = ^, O, - *, (A. 6)\, 

which enables us to calculate step by step all the coefficients C A in 
terms of c 2 and <? 3 , and consequently in terms of g^ and g z , we find 
thus 



,, = ____ 

* 2 4 3 5 2 ' 5 ~2*.5.7 11 ' 

This computation brings out the remarkable algebiaic fact that all 
the sums S'l/(2 w) Zn are expressible as polynomials in terms of the 
first two 

We know a pnori the roots of p'(w). This function, being of the 
thud cider, has thiee roots in each parallelogram of periods Since 
it is odd, it has the roots u = o, u = /, u = &" = o> + ' ( 68, notes). 
By (27) the roots of the equation 4 p 8 <?jp ^ 3 = are precisely 
the values of p () for w = o>, CD', co". These three roots are ordinarily 
represented by e v e^ e^ : 

*i = P ( to ). a = P C* 1 )* 6 s = P ( w ")- 

These three roots are all different ; for if we had, for example, ^ = 2 , 
the equation p(w)= e 1 would have two double roots <o and o> f in the 
interior of a parallelogram of periods, which is impossible, since p(w) 
is of the second order Moreover, we have 



and between the invariants g^ g s and the roots e v e 2 , e 8 we have the 
relations 

e i + 6 * + \ = > e i e 2 + e i e s + W* = - 'f ' e iV* = f 
The discriminant (*^ 27 <7l)/16 is necessarily different from zero. 



71. The function (u). If we integrate the function p(^) 
along any path whatever starting from the origin and not passing 
through any pole, we have the relation 



The series on the right represents a function which is analytic 
except for poles, having all the points u = 2w, except u = 0, for 



160 SINGLE-VALUED ANALYTIC FUNCTIONS [m, 71 

poles of the fiist order Changing the sign and adding the frac- 
tion 1/u, we shall put 

(29) 

The preceding relation can be written 

(30) rT ' % 11 * '" ^ ' 1 



and, taking the derivatives of the two sides, we find 



It is easily seen from either one of these formulae that the function 
(u) is odd. In the neighborhood of the origin we have by (23) 
and (30), 



The function (u) cannot have the periods 2 <o and 2 a/, for it would 
have only one pole of the first order in a parallelogram of periods. 
But since the two functions (u 4- 2 w) and (u) have the same deriva- 
tive p (u), these two functions differ only by a constant , hence the 
function (u) increases by a constant quantity when the argument u 
increases by a period It is easy to obtain an expression for this con- 
stant. Let us wiite, for greater clearness, the formula (30) in the form 



Changing it, to u -f- 2 u> and subtracting the two f ormulse, we find 
C( + 2)-C()= r "p(v)dv. 

Jit 

We shall put 

xiu+20) rM + 2i' 

217= I p(v)dv, 2i/:= / p(v)dv. 

Then 17 and ^ r are constants independent of the lower limit u and of 
the path of integration This last point is evident a priori, since all 
the residues of p(v) are zero. The function (u) satisfies, then, the 
two relations 



f( + 2 ) = {(*) + 2 % {(u + 2 */) = f 

If we put in these formulae w = o> and w = o r respectively, we 
find 17 = (*>), ,/ =(0,'). 



Ill, 71] ELLIPTIC FUNCTION'S 161 

There exists a very simple relation between the four quantities o>, 
a/, iq, iff. To establish it we have only to evaluate in two ways the 
integral J(ii)du, taken along the parallelogram whose vertices are 
U Q , z* -f 2 <o, + 2 (o -f 2 o', w -j- 2 a/, "We shall suppose that () 
has no poles on the boundary, and that the coefficient of i in o>'/<o is 
positive, so that the veitiees will be encountered in the order in 
which they aie written when the boundary of the parallelogiam is 
described in the positive sense There is a single pole of (it) in the 
interior of this boundary, with a residue equal to -f- 1 5 hence the 
integral under consideration is equal to 2 TTI On the other hand, by 
68 the sum of the mtegials taken along the side joining the veitiees 
U Q> U Q -|- 2 <D and along the opposite side is equal to the expiession 



/ 

t/Uft 



- 4:0)77'. 



Similarly, the sum of the integrals coming from the other two sides 
is equal to 4 0/17 We have, then, 

/OO\ f ___ f 2H a" 

which is the relation mentioned above. 
Let us again calculate the definite integral 

"**(*)*>, 

taken along any path whatever not passing through any of the poles 
We have ^ f/ N &/ 



so that F(u) is of the form F(ii) ~2iju + K, the constant K being 
determined except for a multiple of 2 iri y for we can always modify 
the path of integration without changing the extremities in such a 
way as to increase the integral by any multiple whatever of 
To find this constant K let us calculate the definite integral 



along a path, very close to the segment of a straight line which joins 
the two points o> and . This integral is zero, for we can replace the 
path of integration by the rectilinear path, and the elements of 
the new integral cancel in pairs. But, on replacing u by o> in the 
expression which gives F(w), we have 



/vhw 

I ^(v)^ = - 

J 6 



162 SINGLE-VALUED ANALYTIC FUNCTIONS [in, 71 

and since we have also 



we can take K = 2 yu iri Hence, without making any supposition 
as to the path of integiation, we have, in general, 



(33) C" 

Ju 



where m is an integer, and we have an analogous formula for the 
integral f^ +2 ^(v)du 

72. The function CT(M). Integrating the function (w) l/ along 
any path starting from the origin and not passing through any pole, 
we have 



and consequently 

(34) .jrDw-g*- Ml 

The integral function on the right is the simplest of the integral 
functions which have all the periods 2 w for simple roots , it is the 
function <r(z) 



(35) ,()- .n--. 

The equality (34) can be written 

(34') o-() = 

whence, taking the logarithmic derivative of both sides, we obtain 



The function <r(^), being an integral function, cannot be doubly 
periodic. When its aigument increases by a period, it is multiplied 
by an exponential factor, which can be determined as follows : 
From the formula (34*) we have 



cr(u) U 

This factor was calculated in 71, whence we find 
(37) cr(u 



Ill, 73] ELLIPTIC FUNCTIONS 163 

It is easy to establish in a similar manner the relation 
(38) o-O + 2 ') = - eW+"><r(ii). 

From either of the formulae (35) or (34') it follows that <r(w) is 
an odd function 

If we expand this function <r(w) in powers of u, the expansion 
obtained will be valid for the whole plane It is easy to show that 
all the coefficients are polynomials in g Q and g y For we have 



3.4" 5.6" 2X(2X-1) 

<r(^)=we"" r * U *"" r ^ we ~ ". 

We see that there is no term in if and that any coefficient is a 
polynomial in the c A 3 s and theiefoie in the invariants #, and ^ 8 , 
the first five terms are as follows 



. 

" 2 4 35 2*357 2 9 3 2 5.7 2 7 .3 2 5 2 .7.11 

The three functions p(z), ()? *00 are ^he essential elements of 
the theory of elliptic functions The first two can be derived from 
or(w) by means of the two relations (u) = <r'()/<r(w), p(u) = '(M) 

73. General expressions for elliptic functions. Every elliptic function 
f(u) can be expiessed in terms of the single function <r(z), or again 
in terms of the function (u) and of its derivatives, or finally in 
terms of the two functions p(w) and p'(z) We shall present con- 
cisely the three methods. 

Method 1. Expression of f(u) in terms of the function <r(u). Let 
a i> a Q> ' '9 a n ^ e ^ ne zeros ^ tne function f(u) in a parallelogram of 
periods, and l lt 1 9 - , Z> n the poles o/(w) in the same parallelogram, 
each of the zeros and each of the poles being counted as often as is 
required by its degree of multiplicity. Between these zeros and poles 
we have the relation 

(40) r l + a s| +.. +a n = b l + lt+- 

where 2 O is a period 

Let us now consider the function 



This function has the same poles and the same zeros as the function 
f(u), for the only zeros of the factor ar(u a t ) are u = a t and the 



164 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 73 

values of u which differ from a % only by a period On the other hand, 
this function <(M) is doubly penodic, for if we change u to u -f- 2 o>, 
for example, the relation (37) shows that the numerator and the 
denominator of <f>(u) are multiplied lespectively by the two factors 



and these two factors are equal, by (40) Similaily, we find that 
<^(u -h 2o>') = <f>(u). The quotient f(u)/<]> (11) is therefore a doubly 
periodic function of u having no infinite values , that is, it is a 
constant, and we can write 



tl \ r i 

/W- c ^ _ b ^ u _ ^ ^ _ ^ 

To determine the constant C it is sufficient to give to the vaiiable u 
any value which is neither a pole nor a zero 

More generally, to express an elliptic function /() in terms of 
the function <r(u), when we know its poles and its zeios, it will suf- 
fice to choose n zeros (a^, a^ , <) and n poles (l^ b& - , b) in 
such a way that Sd^ = S^^ and that each root of f(u) can be obtained 
by adding a period to one of the quantities af, and each pole by 
adding a period to one of the quantities % These poles and zeros 
may be situated in any way in the plane, provided the preceding 
conditions are satisfied 

Method 2. Expression off(u) in terms of the function and of its 
derivatives. Let us consider k poles a 1? a 2 , -, a L of the function/(w) 
such that every other pole is obtained by adding a period to one 
of them We could take, for example, the poles lying in the same 
parallelogram, but that is not necessary. Let 



(u a t ) w * 

be the principal part of f(u) in the neighborhood of the point a t . 
The difference 



is an analytic function in the whole plane. Moreover, it is a doubly 
periodic function, for when we change u to u 4- 2 o>, this function is 
increased by 2^A^, which is zero, since S^i represents the sum 



HI, 73] ELLIPTIC FUNCTIONS 165 

of the residues in a parallelogiam. That difference is theiefore a 
constant, and we have 



;= C + A>t(u - a.) - Jj,r t H - a,) - - - 
(42) -=I L 

-K-i^^ji^-^ 0] 

The preceding formula is due to Hermite. In order to apply it we 
must know the poles of the elliptic function f(ii) and the corre- 
sponding principal parts Just as formula (41) is the analogon of the 
foimula which expresses a lational function as a quotient of two 
polynomials decomposed into then lineai factors, the formula (42) 
is the analogon of the formula for the decomposition of a rational 
fraction into simple elements. Here the function (u a) plays the 
part of the simple element. 

Method 3 Expression off(u) in terms ofp(i() and ofp'(u) Let 
us consider first an even elliptic function f(u) The zeros of this 
function whith 0,1 e not pet fads, are symmetric in pairs We can 
theiefore find n zeios (a j9 a 2 , , a n ) such that all the zeros except 
the penods are included in the expressions 



We shall take, for example, the parallelogram whose vertices aie 
<o + <o f , a/ o>, a) <>', co <o r and the zeios in this parallelogram 
lying on the same side of a straight line passing through the origin, 
carefully excluding half the boundary in a suitable manner. If a 
zero a t is not a half-period, it will be made to appear in the sequence 
a l9 a 2) , a n as often as there are units in its degree of multiplicity. 
If the zero a v for example, is a half-period, it will be a zero of even 
order 2 r ( 68, notes) We shall make this zero appear only r times 
in the sequence a l9 a 2 , , a n With this understanding, the product 



has the same zeros, with the same orders, as /(*), excepting the case 
of /(O) = Similarly, we shall form another product, 



having the poles of f(u) for its zeros and with the same orders, 
again not considering the end points of any period. Let us put 

CP () ~ P Ml . 



166 SINGLE-VALUED ANALYTIC FUNCTIONS [m, 73 

the quotient /00/<K'0 1S an elliptic function which has a finite 
value different ft oni zem for eveiy value of u which is not a penocl 
This elliptic function i educes to a constant, for it could only have 
periods for poles , and if it did, its leciproeal would not have any 
poles We have, then, 

,/, A _ r EP 00 - P (i)] [P fr ) ~ P K)] CP 00 - P (<Q] . 
/( } "" [POO- PCOTPOO- P&)] [POO- P&.)] 



If / 1 (?/) is an odd elliptic function, ,/iOO/P'OO 1S an ev ^ n function, 
and therefore this quotient is a lational function of p (z) Finally, 
any elliptic function F(tt) is the sum of an even function and an 
odd function . 



Applying the preceding results, we see that every elliptic function 
can be expressed in the form 

(43) J P()= *[>()] + ?'()*,!>()], 

where -R and 72 1 are rational functions 

74 Addition formulae. The addition formula for the function sin x 
enables us to express sin (a 4- Z>) in terms of the values of that func- 
tion and of its derivative for x = a and x = I There exists an 
analogous formula for the function p(w), except that the expression 
for p(u -f- #) in terms of p(^), p(v), p'(w), p'(*0 is somewhat more 
complicated on account of the presence of a denominator 

Let us first apply the general formula (41), in which the function 
cr(u) appears, to the elliptic function p(w)""P(^) ^ e see a ^ once 
that cr(u -I- v) <r(u v)/cr\ii) is an elliptic function with the same 
zeros and the same poles as p (w) p (y) We have, then, 

/ \ /\ ^ < 
p()_p( B )-C 



in order to determine the constant C it suffices to multiply the two 
sides by &*(u) and to let u approach zero We thus find the relation 
1 = Ca*(v), whence we derive 

,... , , ,. <r(u + - 

(44) PW P() = -- _j 

^ / <r\/<r\/ 2 



If we take the logarithmic derivative on both sides, regarding v as 
a constant and u as the independent variable, we find 



Ill, 74] ELLIPTIC FUNCTIONS 167 

or, interchanging u and v in this result, 



Finally, adding these two results, we obtain the relation 



which constitutes the addition foimula for the function (u). 

Differentiating the two sides with respect to u, we should obtain 
the expression for p (u + v) , the right-hand side would contain 
the second derivative p"(w), which would have to be replaced by 
6 p 3 (w) gJ2 This calculation is somewhat long, and we can obtain 
the result in a more elegant way by proving first the relation 



(46) p(u + v) + p(tO + POO = [C(* + *)- COO- COOT- 

Let us always regard u as the independent variable , the two sides 
are elliptic functions having for poles of the second order u = 0, 
u = v, and all the points deducible from them by the addition of 
a period In the neighborhood of the origin we have 

CO* + *)- COO- C00= COO+ POO+ ---- COO- COO 
=- ^ + *C'00+ *+ 

and consequently 

- 2 + . . -. 



The principal part is l/% 2 , as also for the left-hand side. Let us 
compare similarly the principal parts in the neighborhood of the pole 
u = v Putting u = v -f A, we have 



- C(* - ) - COOT- - 2 C'W + - - 

The principal part of the right-hand side of (46) m the neighbor- 
hood of the point u = v is, then, /(u + vf, just as for the left- 
hand side Hence the difference between the two sides of (46) is 
a constant To find this constant, let us compare, for instance, the 
developments in the neighborhood of the origin. We have in this 
neighborhood 

p( + v) + p(i*) 4- p(0 = 



168 SINGLE- VALUED ANALYTIC FUNCTIONS [III, 74 



Comparing this development with that of [(w -f r) (M) (*>)] 2 , 
we see that the difteience is zeio foi it = The i elation (46) is there- 
fore established Combining the two equalities (45) and (46) , we 
obtain the addition foinmla foi the function p(w) 



75. Integration of elliptic functions. Hermite's decomposition for- 
mula (42) lends itself immediately to the integration of an elliptic 
function Applying it, we find 




We see that the integral of an elliptic function is expressible in 
terms of the same transcendentals or, , p as the functions themselves, 
but the function <r(u) may appear in the result as the argument of 
a logarithm In ordei that the integral of an elliptic function may 
be itself an elliptic function, it is necessary first that the integral 
shall not present any logarithmic critical points , that is, all the 
residues A ( f must be zero If this is so, the integral is a function 
analytic except foi poles In order that it be elliptic, it will suffice 
that it is not changed by the addition of a period to u, that is, that 

2?2>-lJ> = 0, 2CV - 2^2^= 0; 



whence we derive C = 0, ^A ( f = 0. If these conditions are satisfied, 
the integral will appeal in the form indicated by Hermite's theorem. 

"When the elliptic function which is to be integrated is expressed in terms 
of p(u) and p'(w) it is often advantageous to start from that form instead of 
employing the general method Suppose that we wish to integrate the elliptic 
function R [p (w)] + p' (u) R l [p (u)], R and R l being rational functions We have 
only to notice in regard to the integral fR l [p (u)] p'(u) du that the change of 
variable p (u) = t reduces it to the integial of a rational function. As for the 
integral fR [p (u)] du, we could reduce it to a certain number of type forms by 
means of rational operations combined with suitably chosen mtegiations by 
parts ; but it turns out that this would amount to making in another form the 
same reductions that were made in Volume I ( 105, 2d ed. , 110, 1st ed.). For, 
if we make the change of variable p (u) = t , which gives 

<#, or dtt dt 



m,fc75] ELLIPTIC FUNCTIONS 169 

the mtegral/JR [p (u)] du takes the form 

E(t)dt 



We have seen how this integral decomposes into a rational function of t and 
of the radical V4 t s g^t g^ a sum of a certain number of integrals of the 
f oim ft n dt/ V4 t s g 2 t </ 3 , and finally a certain number of integrals of the form 

dt 



j 



wheie P() is a polynomial prime to its derivative and also to 4 3 g z t g^ 
and where Q (t) is a polynomial prime to P (t) and of lower degree than P (t) 
Returning to the variable u, we see that the mtegial fR[p(u)]du is equal 
to a rational function of p(ic) and p'(tt), plus a certain number o integrals 
such asJ"[p(u)] n <Zit and a certain number of other integrals of the form 



r 
J 



P &>()] ' 

and this reduction can be accomplished by rational operations (multiplications 
and divisions of polynomials) combined with certain integrations by parts. 

We can easily obtain a recuirent formula for the calculation of the integrals 
I n = J*[p (u)1 n du If, m the i elation 

{[P ()]- WM)} = (-!) [P M]- 2 P' 2 (u) + [p (tt)]- ip"(), 

we replace p /2 (w) and p /7 (w) by 4p 3 (w) gr 2 p(w) gr s and 6p 2 (w) gr 2 /2 
respectively, there results, after arranging with respect to 



and from this we derive, by integrating the two sides, 

/ i\ 
(50) [p(u)'] n ~- 1 p'(u) = (4ra + 2) I n + i in ir 2 J n _i (?i l)gr 3 I B _ 2 

By putting successively % = 1, 2, 3, - m this formula, all the integrals I n 
can be calculated successively from the first two, J = w, Z 1 = f (u) 

To reduce further the integrals of the form (49), it will be necessary to know 
the roots of the polynomial P(t). If we know these roots, we can reduce the 
calculation to that of a certain number of integrals of the form 

du 



where p(v) is different from ^, <? 2 , e 8 , since the polynomial P(t) is "prime to 
4 $8 _ gj _ g The value of u is therefore not a half-period, and p'(v) is not 
zero. The formula 



established in 74, then gives 

(51) C - ^L_^ = ^ 
v ' Jp(w)-p() p'() 



170 SINGLE-VALUED ANALYTIC FUNCTIONS [ill, 76 

76. The function B The series by means of which we have defined the func- 
tions p(w), (u), ff(u) do not easily lend themselves to numerical computation, 
including even the power series development of <r(u), which is valid for the 
whole plane The founders of the theory of elliptic functions, Abel and Jacobi, 
had introduced another remarkable transcendental, which had previously been 
encountered by Fourier in his work on the theory of heat, and which can be 
developed in a very rapidly convergent series , it is called the 6 function We 
shall establish briefly the principal properties of this function, and show how 
the Weierstrass <r (u) function can be easily deduced from it 

Let r = r + si be a complex quantity in which the coefficient a of i is positive 
If v denotes a complex variable, the function 6 (v) is defined by the series 

- +w / ]\a 

(52) 6 (V) =~S (- l)gfV* + *J e fl + I)w q _ girtr^ 

1 ^* 

00 . 

which may be regarded as a Laurent series in which & w has been substituted 
for z This series is absolutely convergent, for the absolute value U n of the 
general term is given by 



if u = a + j8z , hence V U n approaches zero when n becomes infinite through 
positive values, and the same is true of Vt71_ n It follows that the function 
8 (o) is an integral transcendental function of the variable u It is also an odd 
function, for if we unite the terms of the series which correspond to the values 
n and n 1 of the index (where n varies from to +00), the development 
(52) can be replaced by the following formula 

(53) 0(v) = 2 V ( l) n gv + 2) sin(2n+l)7n>, 

o 
which shows that we have 

Q (-fl)=- 0(v), 0(0) = 

When v is increased by unity, the general term of the series (52) is multi- 
plied by e (2+ !>** = - 1 We have, then, 6 (v + 1) = 8 (v) If we change 1? to 
v + T, no simple relation between the two series is immediately seen , but if 
we write 



r) = ] (- l)n 

00 

and then change n to n 1 in this series, the general term of the new series 






is equal to the general term of the series (52) multiplied by q- 1 e-*" w . Hence 
the function Q (v) satisfies the two relations 

(54) 0(t> + l)=-0(t>), 0(t> + r)=-g-i er 2inv0( l ,) 

Since the origin is a root of 6 (t>), these relations show that Q (v) has for zeros all 
the points m^ + m 2 T, where m^ and m 2 are arbitrary positive or negative integers 



III, 76] ELLIPTIC FUXCTIOXS 171 

These are the only roots of the equation (t) = For, let us consider a 
parallelogram whose vertices are the four points v , v -f 1, V Q -f 1 + T, t? + r, 
the fhst veitex V Q being taken m such a -way that no loot of 0(e) lies on the 
boundaiy We shall show that the equation 0(v) = has a single root in this 
parallelogiam For this purpose it is sufficient to calculate the integral 



along its boundary in the positive sense By the hypothesis made upon r, we 
encounter the vertices in the order in which they aie wntten 
From the relations (54) we derive 




.. s 
0<B) 

The fiist of these relations shows that at the coiiespondmg points n and n' 

(Fig 27) of the sides AD, BC, the function 0' ()/#() takes on the same value 

Since these two sides are described in 

contrary senses, the sum of the eor- 

responding integrals is zero On the 

contrary, if we take two corresponding 

points m mf on the sides J.J3, DC, the 

value of 0'0?)/0(o) at the point m' is 

equal to the value of the same function 

at the point m, diminished by %m. The 

sum of the two mtegials coming from FIG 27 

these two sides is therefore equal to 

/(CD) ~~ ^ fl^^i fck at is i to 2 m As there is evidently one and only one point 

in the parallelogram A BCD which is represented by a quantity of the form 

m 1 4- m s r, it follows that the function 6 (v) has no other roots than those found 

above. 

Summing up, the function B (v) is an odd integral function , it has all the 
points m l + m z r for simple zeros , it has no other zeros , and it satisfies the 
relations (54) Let now 2 w, 2 ' be two periods such that the coefficient of i in 
wVo> is positive In 9 (a>) let us replace the variable V by ie/2 w and r by w'/w, 
and let < (w) be the function 

(55) *(> = 



Then $ (u) is an odd integral function having all the periods 2 w = 2 mw -f 

for zeros of the first order, and the relations (54) are replaced by the following 



(56) 

These properties are very nearly those of the function <r (u) In order to re- 
duce it to ff (u), it suffices to multiply # (w) by an exponential factor. Let us put 

(57) Vfa) = ^^%(M), 

where 57 is the function of w and w' defined as m 71. This new function ^ (u) 
is an odd integral function having the same zeros as (M) The first of the 



172 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 76 

relations (56) becomes 

(58) f(ii + a)=-^5= fII+9li %W=-e 1 ^ + - ) *W- 

We have next 



or, since i?t/ i?' = iri/2, 

(59) I (u + 2 ') = - e 2 *< + >'> ^ (M) 

The relations (58) and (59) aie identical with the relations established above 
for the function <r(u) Hence the quotient t(u)/<r(u) has the two penods 2w 
and 2 u', for the two teims of this latio aie multiplied by the same factor when 
u increases by a period Since the t\vo functions have the same zeros, this 
quotient is constant , moieover, the coefficient of u in each of the two develop- 
ments is equal to unity We have, then, a- (u) = $ (u), 01 

(60) , ( 

and the function <r(u) is expressed in terms of the function 0, as we proposed 
If we give the argument v real values, the absolute value of q being less than 
unity, the senes (53) is rapidly convergent We shall not further elaboiate 
these indications, which suffice to suggest the fundamental part taken by the 
6 function m the applications of elliptic functions 

III INVERSE FUNCTIONS CURVES OF DEFICIENCY ONE 

77. Relations between the periods and the invariants. To every 
system of two complex numbers w, <>', whose ratio o>'/a> is not real, 
corresponds a completely determined elliptic function p(u), which 
has the two periods 2 o>, 2 a>', and which is regular for all the values 
of u that are not of the foim 2 mo> + 2 w V, all of which are poles of 
the second order The functions (u) and <r(u) 9 which are deducible 
fiom p(u) by one or by two integrations, respectively, are likewise 
determined by the system of periods (2 <o, 2 <o f ). When there is any 
reason for indicating the penods, we shall make use of the notation 
P(K|O, *>'), (w|> <> f )? <r(w|> <>') to denote the three fundamental 
functions. 

But it is to be noticed that we can replace the system (w, o> f ) by 
an infinite number of other systems (O, O 1 ) without changing the 
function p(w). For let m, m f , n, n 1 be any four positive or negative 
integers such that we have mn' m'n = 1 If we put 



we shall have, conversely, 

<u =s (n'Q %O') 3 o)' =3 (mO' m'O), 



Ill, 77] INVERSE FUNCTIONS 173 

and it is clear that all the periods of the elliptic function p(*f) are 
combinations of the two periods 2Q, 2Q', as well as of the two 
periods 2 <o, 2 >' The two systems of periods (2 <D, 2 *>') and (2 O, 2 O') 
are said to be equivalent The function p(|Q, G') has the same 
periods and the same poles, with the same principal parts, as the 
function p(w|<o, o>'), and their difference is zeio for u = 0. They are 
therefore identical This fact results also fiorn the development 
(22), for the set of quantities 2 m<* + 2 ??zV is identical with the 
set of quantities 2mO-j-2w'Q'. For the same reason, we have 
i(w|O, O')= 0jo>, >') and <T(H|Q, O') = <r(|o>, o/) 

Similarly, the three functions p(w), (?<), <r(z*) are completely deter- 
mined by the invariants ^ 2 , ^ 3 For we have seen that the function 
<r (u) is represented by a power-series development all of whose coeffi- 
cients are polynomials in <? , g^ We have, then, (?/) = o- f (i^)/<r (M), 
and finally p(w) = '(w). In older to indicate the functions which 
coriespond to the invanants </, and g$ we shall use the notation 

P ( u : ff <7 8 )> > ^ ^s)? "C^ 5 &, ^)- 

Just here an essential question piesents itself. While it is evi- 
dent, from the veiy definition of the function p (?;), that to a system 
(o>, <o f ) corresponds an elliptic function p(w), provided the ratio 
o)'/<o is not real, there is nothing to prove a priori that to every 
system of values for the invariants g^ g z corresponds an elliptic 
function. "We know, indeed, that the expression g\ 27 #f must be 
different from zero, but it is not certain that this condition is suffi- 
cient The problem which must be treated here amounts in the end 
to solving the transcendental equations established above, 



(61) fc- 60 2'(2i + 2 f l ) ' ^"-"^(2 

for the unknowns o>, o> r , or at least to determining whether or not 
these equations have a system of solutions such that co'/o is not xeal 
whenever g| 27 g\ is not zero If there exists a single system of solu- 
tions, there exist an infinite number of systems, but there appears 
to be no way of approach for a direct study of the preceding equations 
We can ainve at the solution of this problem in an indirect way by 
studying the inversion of the elliptic integral of the first kind 

Note. Let , <*' be two complex numbers such that U'/QJ is not real. The corre- 
sponding function p (u | , /) satisfies the differential equation 



174 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 77 

where g 2 and g s are defined by the equations (61) Por u = w, p (w) is equal to 
one of the roots e x of the equation 4p 8 2 p g z = When w varies from 
to w, p(it) describes a cuive L going fiom infinity to the point e l From the 
relation du = dp/V4 p 8 # 2 p ^ 3 we conclude that the half -period w is equal 
to the definite integral 



taken along the curve L An analogous expression for w' can be obtained by 
replacing e l by e 3 in the preceding integial 

We have thus the two half -periods expressed m terms of the invanants g^ <7 8 
In order to be able to deduce from this result the solution of the problem before 
us, it would be necessary to show that the new system is equivalent to the system 
(61), that is, that it defines g% and g z as single-valued functions of w, w'. 

78. The inverse function to the elliptic integral of the first kind. Let 
R(z) be a polynomial of the third or of the fourth degiee which is 
prime to its derivative. We shall write this polynomial in the form 



where a v a# a^ # 4 denote four different roots if R(z) is of the 
fourth degree. On the other hand, if R (z) is of the third degree, we 
shall denote its three roots by a is a a , & 8 , and we shall also set <z 4 =oo, 
agreeing to replace oo by unity in the expression R (&) 
The elliptic integral of the first kind is of the form 



where the lower limit # is supposed, for definiteness, to be different 
from any of the roots of R (#) and to be finite, and where the radical 
has an assigned initial value. If R () is of the fourth degree, the 
radical VjR(a) has four critical points a v a z , 8 , a 4 , and each of the 
determinations of ^/R(z) has the point & = oo for a pole of the second 
order. If R (z) is of the third degree, the radical VjR(js) has only 
three critical points in the finite plane a l} a, z a 8 , but if the variable 
z describes a circle containing the three points a l} a z , a s , the two 
values of the radical are permuted The point z = oo is therefore a 
branch point for the function V-R (s). 

Let us recall the properties of the elliptic integral u proved in 
55. If u(z) denotes one of the values of that integral when we 
go from the point to the point & by a determined path, the same 
integral can take on at the same point # an infinite number of deter- 
minations which are included in the expressions 



(63) u = u(z)+2mv> + 2mV, u / u(z) + 2m<*> 



Ill, 78] INVERSE FUNCTIONS 175 

if the path is varied In these foimulse m and w'are two entirely 
arbitraiy integers, 2 o> and 2o>' two periods whose ratio is not leal, 
and 7 a constant which we may take equal, for example, to the 
integral over the loop described about the point ^ 

Let p (u | <D, ft)') be the elliptic function constructed with the periods 
2 a), 2 a/ of the elliptic integral (62). Let us substitute in that func- 
tion for the vanable u the integral (62) itself diminished by 7/2, 
and let (z) be the function thus obtained 

dz I >f l / M _ w 
>J *>( u 2"' 




(64) ^ v ~, ^ , , /^-rr % 



This function < (z) is a single-valued function of z In fact, if we 
leplace u by any one of the determinations (63), we find always, 
whatever m and m' may be, 



or 



which shows that <& (3) is single-valued 

Let us see what points can be singular points for this function 
& (z) First let x be any finite value of * different from a branch 
point. Let us suppose that we go fiom the point to the point z l 
by a definite path We arrive at z 1 with a ceitain value for the 
radical and a value u t for the integral In the neighborhood of the 
point z v 1/VJR() is an analytic function of z, and we have a 
development of the form 



Whence we derive 

(65) : 

If w t 7/2 is not equal to a period, the function p (u 7/2) is 
analytic in the neighborhood of the point u l9 and consequently <f (s) 
is analytic in the neighborhood of the point z r If ?/ t 7/2 is a 
period, the point u^ is a pole of the second order for p(u 7/2), and 
therefore * x is a pole of the second order for *(*), for in the neigh- 
borhood of the pomt u^ 



where P is an analytic function. 



176 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 78 

Suppose next that 2 approaches a cutical point a t In the neigh- 
borhood of the point a z we have 



where P t is analytic for z = a l} or 




whence, integrating term by teim, we find 
(66) u = it. 

If K, 1/2 is not a penod, p(u 1/2) is an analytic function of w 
in the neighboihood of the point w t Substituting in the develop- 
ment of this function in poweis of u v t the value of the difference 
u _ Vj obtained fiom the foimula (66), the fractional powers of 
(z _ # t ) must disappear, since we know that the left-hand side is a 
single-valued function of , hence the function $() is analytic in 
the neighboihood of the point a l Let us notice in passing that this 
shows that w t 1/2 must be a half-period Similarly, if w, 1/2 is 
equal to a period, the point a t is a pole of the first order for $(#) 

Finally, let us study the function $(z) for infinite values of z 
We have to distinguish two cases accoidmg as R(s) is of the fourth 
degiee or of the thud degree. If the polynomial R (z) is of the fouith 
degree, exterior to a circle C described about the origin as center and 
containing the four roots, each of the determinations of 1/Vj? (2) is 
an analytic function of l/ For example, we have for one of them 



and it would suffice to change all the signs to obtain the develop- 
ment of the second determination. If the absolute value of becomes 
infinite, the radical 1/VjJ (*) having the value which we have ]ust 
written, the integral approaches a finite value ^ w , and we have in 
the neighborhood of the point at infinity 

- 5-ft-ft- - 

If u^ 1/2 is not a period, the function p(u 1/2) is regular for 
the point u^ and consequently the point 2 = oo is an ordinary point 
for $ (z). If u n 1/2 is a period, the point . is a pole of the second 



m, 78] INVERSE FUNCTIONS 177 

order for p (u 1/2), and since we can write, in the neighborhood of 
the point z = oo , 



the point & = oo is also a pole of the second ordei for the function <b(z) 
If R(z) is of the third degree, we have a development of the form 



which holds exterior to a circle having the ongin for center and 
containing the three critical points a v 2 , 8 It follows that 

(68) u = u m 

Reasoning as above, we see that the point at infinity is an ordi- 
nary point or a pole of the fiist oider for &(z). The function $(s) 
has certainly only poles for singulai points , it is therefore a rational 
function ofz } and the elliptic integral of the fiist kind (62) satisfies 
a relation of the form 

(69) 

where <E> (z) is a rational function We do not know as yet the degree 
of this function, but we shall show that it is equal to unity. For 
that purpose we shall study the inveise function In other words, 
we shall now consider u as the independent variable, and we shall 
examine the properties of the upper limit z of the integral (62), con- 
sidered as a function of that integral u We shall divide the study, 
which requires considerable care, into several parts 

1) To every finite value of u correspond m values ofz if m is the 
degree of the rational function & (z) 

Tor let u^ be a finite value of u The equation $ (z) = p (u t 1/2) 
determines m values for z, which are in general distinct and finite, 
though it is possible for some of the roots to coincide or become 
infinite for particular values of u^ Let z l be one of these values 
of z The values of the elliptic integral u which correspond to this 
value of z satisfy the equation 



we have, then, one of the two relations 
u = u + 2m + 2m<', u = I- 



178 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 78 

In eithei ease we can make the vanable describe a path from # to 
gj such that the value of the integial taken ovei this path shall be 
precisely 2/ r If the function &(z) is of degree m, there aie then m 
values of z for which the integial (62) takes a given value u. 

2) Let 11 ^ be a finite value of u to which corresponds a finite value 
^ of z , that determination, ofz which approaches z 1 when u approaches 
ii l is an analytic function of u in the neighborhood of the point u^ 

For if z l is not a cutical point, the values of u and z which ap- 
proach respectively u t and z l are connected by the relation (65), where 
the coefficient a Q is not zero By the general theorem on implicit 
functions (I, 193, 2d ed , 187, 1st ed ) we deduce from it a 
development for & z 1 in positive integral powers of u u r 

If, for the particular value u^ z were equal to the critical value # t , 
we could in the same way consider the right-hand side of (66) as a 
development in poweis of Vs a t Since a is not zero, we can 
solve (66) for V# a,, and therefore for z a l9 expressing each of 
them as a power series in u 7/,. 

3) Let u^ be one of the values which the integral u takes on when 
1 2 1 becomes infinite , the point w w is a pole for that determination ofz 
whose absolute value becomes infinite 

In fact, the value of the integral u which approaches u^ is repre- 
sented in the neighborhood of the point at infinity by one of the 
developments (67) and (68) In the fiist case we obtain for 1/z a 
development in a series of positive powers of u u^ 

i = j S l (*- < )+ j 8 i ( tt _iO*+ , A^O; 

in the second case we have a similar development for 1/VS, and 
therefore . 



The point w is therefore a pole of the first or second order for z, 
according as the polynomial R(z) is of the fourth or of the third 
degree 

4) We are going to show finally that to a value ofu there can cor- 
respond only one value of z. Tor let us suppose that as the variable z 
describes two paths going from # to two different points z v z s , the 
two values of the integral taken over these two paths are equal It 
would then be possible to find a path L joining these two points z v # 2 
such that the integral 



HI, 78] INVERSE FUNCTIONS 179 

would be zeio If we represent the integral n = X + Yi by tlie point 
with, the cooidmates (A", Y) m the system of rectangulai axes OX 9 
OF, we see that the point u would describe a closed cuive F when 
the point s descubes the open cmve L. We shall show that this is 
not consistent with the propeities which we have ]ust demonstrated. 
To each value of u theie coi respond, by means of the relation 
p (11 1/2) = $ (z), a finite number of values of , each of which 
vanes in a continuous mannei with it, provided the path described 
by u does not pass through any of the points corresponding to the 
value & = oo * Accoiding to our supposition, when the variable u 
describes in its plane the closed curve T starting fiom the point 
A (U Q ) and returning to that point, describes an open aic of a con- 
tinuous cuive passing from the point ^ to the point # 2 Let us take 
two points Jl/and P (Pig. 28) on the curve r. 
Let the initial value of & at A be % v and let 
#', a" be the values obtained when we reach 
the points AT and P' respectively, after u has 
described the paths AM and A MNP. Again, 
let %" be the value with which we airive at 
the point P after u has described the arc 
AQP It lesults from the hypothesis that 
n and 2" are different Let us join the two 
points M and P by a transversal MP interior to the curve P 7 and let 
us suppose that the variable u describes the aic .eb/iJf and then the 
transversal MP , let i f be the value with which we arrive at the 
point P This value " W1 ^ be different fiom &" or else from #{'. If 
it is different from z[', the two paths AmMP and AQP do not lead 
to the same value of & at the point P If s' f and aj are different, the 
two paths AmMP and AmMNP do not lead to the same value at P ; 
therefore, if we start from the point M with the value z? for , we 
obtain different values for s according as we proceed from M to P 
along the path MP or along the path MNP In either case we see 
that we can replace the closed boundary r by a smaller closed bound- 
ary P t , partly interior to r, such that, when u describes this closed 
boundary, & describes an open arc. Repeating this same operation on 
the boundary T l9 and continuing thus indefinitely, we should obtain 
an unlimited sequence of closed boundaries r, T v T 2 , * having the 
same property as the closed boundary r, Since we evidently can 

* We assume the properties of implicit functions wbich will be established later 
(Chapter V) 




180 SINGLE-VALUED ANALYTIC FUNCTIONS [m, 78 

make the dimensions of these successive boundaiies appioach zeio, 
we may conclude that the boimdaiy T n appioaches a limit point X 
Prom the way in which this point lias been defined, there will always 
exist in the mtenoi of a ciicle of ladms e. described about X as a 
center a closed path not leading the vaiiable z back to its ongmal 
value, however small e may be Now that is impossible, for the point 
X is an oidinaiy point or a pole for each of the different determina- 
tions of 2 , in both cases z is a single-valued function of it, in the 
neighborhood of X We aie thus led to a contradiction in supposing 
that the mtegial fdz/'Vfi (2), taken over an open path L, can be zeio, 
or, what amounts to the same thing, by supposing that to a value of 
it coi res pond two values of z. 

We have noticed above that, if for two different values of # we have 
$ (z^) = & (#), we can find a path L from ^ to # 2 such that the integral 



- 



will be zero Hence the rational function <E> (z) cannot take on the same 
value foi two different values of z , that is, the function (z) must be 
of the first degree (z) = (az + b}/(cz -f d) It follows, fiom the 
relation (69), that 

I 
(70) * = - 



and we may state the following impoitant proposition : The upper 
limit & of an elliptic integral of the first kind, considered as a function 
of that integral, is an elliptic function of the second order 

Elliptic integrals had been studied in a thorough manner by 
Legendie, but it was by reversing the problem that Abel and 
Jacobi were led to the discovery of elliptic functions 

The actual determination of the elliptic function z=zf(ii) con- 
stitutes the problem of inversion By the relation (62) we have 

d* 



and therefore V72(s)=/ f (^). It is clear that the radical VTZfe) is 
itself an elliptic function of u We can restate all the preceding 
results in geometric language as follows 

Let R (z) be a polynomial of the third or fourth degree, prime to its 
derivative ; the coordinates of any point of the curve C 9 



Ill, 78] INVERSE FUNCTIONS 181 

(71) if = K(x), 

can be expressed in terms of elliptic functions of the integral of the 
first kind. 

J ; r x dx r* 

u= / = / 

J Xo y J Xo 

in such a way that to a point (x, y) of that curve corresponds only 
one value of u, any period being disregarded. 

To prove the last pait of the pioposition, we need only remark 
that all the values of u which coriespond to a given value of x aie 
included in the two expressions 




+ 2 MjO) + 2 m 2 a/, I -f- 2 m l <*> + 2 

All the values of u included in the first expression come from an 
even nurnbei of loops descubed about critical points, followed by 
the direct path, fiom X Q to x, with the same initial value of the 
radical V/2(ie) The values of u included in the second expression 
come from an odd number of loops described about the critical points, 
followed by the diiect path fiom X Q to x, wheie the corresponding 
initial value of the ladical ^/R (x) is the negative of the foimer If 
we aie given both x and y at the same time, the corresponding 
values are then included in a single one of the two formulae 

Fiom the investigation above, it follows that the elliptic function 
x =/(w) has a pole of the second order in a parallelogram if R(x) 
is of the third degiee, and two simple poles if E (x) is of the fourth 
degree , hence y =/ / (*0 1S ^ the third or of the fourth order, accord- 
ing to the degree of the polynomial R (ar) 

Note Suppose that, by any means whatever, the coordinates (x, y) 
of a point of the curve y*=tR(x) have been expressed as elliptic 
functions of a parameter v, say x = < (), y = ^(v). The integral of 
the first kind u becomes, then, 



The elliptic function f (tO/^i(*0 cannot have a pole, since u must 
always have a finite value for every finite value of v ; it reduces, 
then, to a constant A, and we have u = hv 4- 2 The constant I 
evidently depends on the value chosen for the lower limit of the 
integral u The coefficient k can be determined by giving to v a 
particular value. 



182 



SINGLE-VALUED ANALYTIC FUNCTIONS [III, 79 



79 A new definition of p(u) by means of the invariants. It is now 

quite easy to answer the question proposed in T7 Given two num- 
bers g 2> g s such that g$ 27^1 is not zero, there always exists an 
elliptic function p(w) for which g^ and ^ are the invariants 
Tor the polynomial 

-R(s)=4**-0V* $s 

is prime to its derivative, and the elliptic integral /<fe/V.R(s) has 
two periods, 2 o>, 2 ', whose ratio is imaginary. Let p (u , <J) be the 
eoi responding elliptic function We shall substitute for the aigu- 
rnent u in this function the integial 



(72) 




-JT, 



where H is a constant chosen in such a way that one of the values 
of u shall be equal to zero for s = o> We shall take II, for example, 
equal to the value of the integial f^d/^/R (c) taken over a ray L 

starting at # We shall show fiist that 
the function thus obtained is a single- 
valued analytic function of z. Let & be 
any point of the plane, and let us denote 
by v and v* the values of the integrals 





starting with the same initial value for 
-*jR(z) and taken over the two paths 
zjriz, zjiz, which together form a closed 
curve containing the three critical points 
of the radical Consider the closed curve 



FIG. 29 



formed by the curve z^mznz^ the segment ^, the circle C of very 
large radius, and the segment Zz^ The function 1/V.ft (z) is analytic 
in the interior of this boundary, and we have the relation 



, i 

JL. tf _ 




which becomes, as the radius of the circle C becomes infinite, 



Hi, 79] INVERSE FUNCTIONS 183 

The values of u resulting from the two paths z Q ?nz, z Q nz theiefore 
satisfy the relation u + u' = 0. From this we conclude that the 
function 




is a single-valued function of z We have seen that it is a linear 
function of the form (az -{- ft) /(ess -{- d) To determine a, 1>, c, d it 
will suffice to study the development of this function in the neigh- 
"boihood of the point at infinity We have in this neighborhood 



/*() 2z^~ 4 - 2 4* 1 / 2*J 16*1 ' 

hence the value of u> which is zero for z infinite, is represented by 
the series 



whence 



It follows that the difference p(ju) s is zeio f or = oo But the 
difference (az -f- V)/(cz> + d) z can be zero for z = oo only if we 
have c = ? Z> = 0, a = d, and the function p(w| o>, <o r ) reduces to 
when we substitute for u the integral (72) Taking the point at 
infinity itself for the lower limit, this integral can also be written in 
the form 

(72() M 

' 

and this relation makes p (u) = #, where the function p (&) is con- 
structed with the periods 2 o> 3 2 o> r of the integral fdz/"VR (z). 

Comparing the values of du/dz deduced from these relations, we 
have p'(w) = VjR(), or, after squaring both sides, 



(73) p(*) = *(*) 

The numbers ^ 2 , g^ therefore, are the invanants of the elliptic func- 
tion p(u), constructed with the periods 2 o>, 2 <o'. This result answers 
the question proposed above in 77 If g% 27 g\ is not zero, the 
equations (61) are satisfied by an infinite number of systems of values 
for <o, o) r . If e v e# 8 are the three roots of the equation 



184 SINGLE-TALUED ANALYTIC FUNCTIONS [III, 79 

one system of solutions is given, for example, by the formulae 

, 

(74) 0, 

v ; 




from which all other systems will be deducible, as has been explained 

In the applications of analysis m which elliptic functions occui, the function 
p (u) is u&ually defined hy its invariants In ordei to carry through the numerical 
computations, it ib necessary to calculate a pair of periods, knowing g 2 and gr 3 , 
and also to be able to find a root of the equation p (u) = A, where A is a given 
constant Poi the details of the methods to be followed, and for information 
regaidmg the use of tables, we can only lefei the leadei to special treatises * 

80. Application to cubics in a plane. When pf 27 g\ is not zero, 
the equation 

(75) f = x*-g^-~g z 

repiesents a cubic without double points This equation is satisfied 
by putting x = p(w), y = p'(w)> wheie the invariants of the function 
p(u) aie piecisely g 2 and g^ To each point of the cubic coi responds a 
single value of u in a suitable paiallelogiam of periods For the equa- 
tion p (M) = x has two roots x and u z in a parallelogram of periods, 
the sum u^ + it 2 is a period, and the two values p'(u ) and p'(^ 2 ) are 
the negatives of each other They are therefore equal lespectively 
to the two values of y which correspond to the same value of x 

In general, the coordinates of a point of a plane cubic without 
double points can be expressed by elliptic functions of a parameter. 
We know, in fact, that the equation of a cubic can be reduced to 
the form (75) by means of a projective transfoirnation, but this 
transformation cannot be effected unless we know a point of inflec- 
tion of the cubic, and the determination of the points of inflections 
depend upon the solution of a ninth-degree equation of a special 
form We shall now show that the parametric representation of a 
cubic by means of elliptic functions of a parameter can be obtained 
without having to solve any equation, provided that we know the 
cooidmates of a point of the cubic 

Suppose first that the equation of the cubic is of the form 

(76) f = a o o* + 86^ + 3 b 2 x + ft,, 

* The formulae (39) which give the development of <r (u) in a power series, and 
those which result from it by differentiation, enable us, at least theoretically, to 
calculate <r (u) , <r'(w) , <r"(u) , and consequently (u) and p (u) , for all systems of values 



Ill, 80] INVERSE FUNCTIONS 185 

m which case the point at infinity is a point of inflection This 
equation can be reduced to the preceding foiin by putting y = - 
x = &!/&() + 4xy& , which gives 



where the invariants g 2 , g z are given by the formulse 



Hence we obtain for the coordinates of a point of the cubic (76) 
the following formulse 



Let us now consider a cubic C^ and let (or, /?) be the coordinates 
of a point of that cubic The tangent to the cubic at this point (#, ft) 
meets the cubic at a second point (', )8 f ) whose coordinates can be 
obtained rationally If the point (or', ') is taken as origin of coor- 
dinates, the equation of the cubic is of the form 



ttfa y) + h 

where < t (x 5 y) denotes a homogeneous polynomial of the i th degree 
(L = 1, 2, 3) Let us cut the cubic by the secant y = for , then a; is 
determined by an equation of the second degiee, 



^C 1 . 0+ *i(li 3= 
whence we obtain 



where R(t) denotes the polynomial <#j(lj 4 < 8 (1, t} ^(1, ), which 
is in general of the fourth degree The roots of this polynomial are 
precisely the slopes of the tangents to the cubic which pass through 
the oiigm * We know a priori one root of this polynomial, the slope 
of the straight line which joins the origin to the point (a, fi). Putting 
t = + 1/t', we find 



where the polynomial Rft) is now only of the third degree. The 
coordinates (#, y) of a point of the cubic C 8 are therefore expressible 
rationally in terms of a parameter t 1 and of the square root of a 

*Two roots cannot be equal (see Vol I, 103, 2d ed , 108, 1st ed ) -TRANS, 



186 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 80 

polynomial R,(t') of the third degree We have ]ust seen how to 
express t' and Vjf? 1 (^') as elliptic functions of a parameter u , hence 
we can expiess x and y also as elliptic functions of u 

It follows fioin the nature of the methods used above that to a 
point (r, y) of the cubic coirespond a single value of t and a definite 
value of -vR(), and hence completely detei mined values of t 1 and 
V-R^tf 1 ) Now to each system of values of t 1 and VTt^') corre- 
sponds only one value of w in a suitable parallelogiam of peiiods, as 
we have already pointed out The expiessions x =f(ii), y =f 1 (u), 
obtained for the cooidinates of a point of C 3 , are therefore such 
that all the determinations of u which give the same point of the 
cubic can be obtained from any one of them by adding to it various 
periods. 

This parametric repiesentation of plane cubics by means of elliptic functions 
is very important * As an example we shall show how it enables us to deter- 
mine the points of inflection Let the expressions for the coordinates be 
x =/(u), y =,/i(tt) , the arguments of the points of intersections of the cubic 
with the straight line Ax -f By + C = are the roots of the equation 



Since to a point (x, y) corresponds only one value of u in a parallelogram of 
periods, it follows that the elliptic function Af(u) + Bf^u) + C must be, in 
general, of the third order The poles of that function are evidently independent 
of -d, B, C , hence if u v u z , u s are the three arguments corresponding respec- 
tively to the three points of intersections of the cubic and the straight line, we 
must have, by 68, 

u 



where X is the sum of the poles in a parallelogram Replacing u by JBT/3 + u 
in/(w) and /^w), the relation can be written m the simpler form 

% + M 2 + u 3 = period 

Conversely, this condition Is sufficient to insure that the three points M l (u=w 1 ), 
If 2 (u = w 2 ), Jf 8 (u = u 3 ) on the cubic shall lie on a straight line For let M'% be 
the third point of intersection of the straight line Jf x J9f s with the cubic, and u$ 
the corresponding argument Since the sum u 1 +u 2 + u% is equal to a period, 
tCj and u differ only by a period, and consequently M'% coincides with M 8 

If u is the value of the parameter at a point of inflection, the tangent at that 
point meets the curve in three coincident points, and 3w must be equal to a 
period We must have, then, u = (2m 1 w + ^m^^/B All the points of inflec- 
tion can be obtained by giving to the integers m l and m a the values 0, 1, 2. 
Hence there are mne points of inflections. The straight line which passes through 

*CLEBSCH, Ueber dtejent&en Curven, deren Coordinatensich als elhptwche JFVnc- 
twnen e^nes Parameters darstelten lassen (Crelfe's Journal, Yol 



Ill, 81] INVERSE FUNCTIONS 187 

the two points of inflection (Zm^w + 2m z b>')/3 and (2mj[w 4- 2m2w')/3 meets 
the cubic in a third point whose argument, 

^u -f- 2(m 2 + mgX 



3 

is again one thud of a period, that is, in a new point of inflection The number 
of straight lines which meet the cubic in three points of inflection is theiefore 
equal to (9 8)/(3 2), that is, to twelve 

Note The points of mteisection of the standard cubic (75) with the straight 
line y = mz -f n are given by the equation p'(u) mp (u) n = 0, the left-hand 
side of which has a pole of the thud older at the point u = The sum of the 
arguments of the points of intersection is then equal to a period If u and u z 
are the aiguments of two of these points, we can take w t u z for the argu- 
ment of the third point of intersection, and the abscissas of these three points 
are respectively p (uj, p (u 2 ), p (w t + U Q ) We can deduce from this a new proof 
of the addition formula for p (u) In tact, the abscissas of the points of inter- 
section are roots of the equation 

4x 3 - gr 2 oj - 3 = (ma? + n) z , 
hence 

m 2 
x t + OJ 2 + JC 3 = p(u l ) + p(ua) + pK + u a ) = ~ 

On the other hand, from the straight line passing through the two points Jf^t^), 
M 2 (w 2 ), we have the two relations p'(u^=mp (u : ) + n, p'(u%) =mp (u 2 ) + w, whence 



and this leads to the relation already found in 74, 



81. General formulae for parameter representation. Let R (x) be a 

polynomial of the fourth degree pi line to its derivative. Considei 
the curve C 4 repiesented by the equation 

(77) f = ^(a;)= a x* + 4a t a; 8 + 6^8? + 4a t as -f a^ 

We shall show how the coordinates x and y of a point of this curve 
can be expressed as elliptic functions of a parameter. If we know a 
root a of the equation R (x) = 0, we have already seen in the treat- 
ment of cubics how to proceed. Putting x = a + 1/x' 9 the relation 
(77) becomes 



where Rfa') is a polynomial of the third degree. Hence the curve (7 4 , 
by means of the relations x = a -j- I/a; 1 , y = 2/ /' 2 , corresponds point for 



188 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 81 

point to the curve Cg of the third degree whose equation is y^Rfa'} 
Now x' and y } can be expiessed by means of a parameter u, in the foim 
x 1 = <rp(w)4-/3, if = ccp l (ii), by a suitable choice of a, /3 and of the 
invariants of p(n) We deduce fiom these relations the following 
expressions for x and y : 



whence we find <7tf = dr/y, so that the parameter u is identi- 
cal, except for sign, with the integral of the fiist kind, fdx/^/R (x), 
and the formulae (78) constitute a generalization of the results for 
the simple case of parametric representation in 80 

Let us considei now the general case in which we do not know any 
loot of the equation J2(;r) = We are going to show that x and y 
can be expressed rationally in terms of an elliptic function p(u) with 
known invariants, and of its derivative p'(&), without introducing any 
other irrationality than a square root. Let us replace for the moment 
x and y by t and v icspectively, so that the relation (77) becomes 

(77') v* = R (t) = a^ + 4 a t f + 6 aj? + 4 a z t + a^ 
The polynomial R (t) can be expressed in the form 



in an infinite numbei of ways, where < 1? < 2 , < 3 aie polynomials of 
the degrees indicated by then subscripts. For let (a, /$) be the cooi- 
dinates of any point on the curve C 4 Let us take a polynomial <f> 2 (t) 
such that 4> 2 (a) = /3, which can be done in an infinite number of ways ; 
then the eqnafaon 



will have the root t = a, and we can put ^(tf) = t a The poly- 
nomial R (t) having been put in the preceding f oim, let us consider 
the auxiliary cubic C 8 represented by the equation 

(79) 

If we cut this cubic by the secant y = tx, the abscissas of the two 
variable points of intersection are roots of the equation 



and can be expressed in the form 



Ill, 81] INVERSE FUNCTIONS 189 

where v is determined by the equation (77') Conversely, we see that 
t and v can be expiessed lationally in terms of the coordinates a, y 
of a point of C 8 by the equations 

(80) ,.J. .- 

Now cc and y can be expressed as elliptic functions of a parameter , 
since we know a point on the cubic <? 3 that is the ongin Then t 
and v can also be expiessed as elliptic functions of u The method is 
evidently susceptible of a great many vai rations, and we have mtio- 
duced only the irrational ft = V/2 (a), wheie a is aibitiaiy. 

We are going to carry thiough the actual calculation, supposing, 
as is always admissible, that we have first made the coefficient a t of t* 
disappeai in R(). We can then write 

a Q R (t) = (a/) 2 + 6 V / + 4 a Q a & t + a a 4 
and put 

^(0 = -!, < 2 (0=V 2 > ^(O^K^ + 'KV + VV 
The auxiliary cubic C g has the form 

(81) 6 a a 2 itf + 4 a^aPy + a^af + 2 a Q f - r = 0. 

Following the general method, let us cut this cubic with the 
secant y~tx, the equation obtained can be wntten in the form 



- 2 a/ - (6 vr/ + 4 a a,# + a A ) = 5 

\J!J/ <lt 

whence we obtain 

i = a *+VV20). 

Conversely, we can express # and V 22 (#) in terms of a; and ^: 

(82) * = J, V^(0 = ^- 

On the other hand, solving the equation (81) for y, we have 



- 2 g Q a s r 2 + V4 gajx 4 - x(a a 4 x 2 -1) (6 g a> g g + 2 Q 



The polynomial under the radical has the root a; = 0. Applying the 
method explained above, we can then express x and y as elliptic 
functions of a parameter. Doing so, we obtain the results 



190 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 81 



where the mvaiiants #,, <7 8 of the elliptic function p(u) have the 
following values 



(84) ?*=-a> ' ff ' = 

Substituting the preceding values for x and y in the expressions 
(82), we find 



(85) 



'-2 



=vz 



*P()-?-7 






We can write these results in a somewhat simpler form by noting 
that the relations 



(86) 







are compatible accoiding to the values (84) of the invariants g^ and g^ 
On the other hand, we can substitute for 



its equivalent p (u +_v)+ p (u) + p (v). Combining these results and 
replacing # and Vj(tf) bj a? and y respectively, we may formulate 
the result in the following proposition . 

The coordinates (x, y) of any point on the curve C 4 , represented by 
the equation (77) (ivhere a^ = 0), can be expressed in terms of a varir 
able parameter u by the formuUe 



(87) 



where the invariants g^ and g^have the values given by the relations (84), 
and where p(t?), p'(v) are determined by the compatible equations (86) 

!From the formula (45), established above ( 74), we derive, by 
differentiating the two sides of that equality, 

1 d 



Ill, 82] INVERSE FUNCTIONS 191 



that is, dx/du = y/^fa^ or du = [Vaj/y] f7x. The parameter it, there- 
fore, represents the elliptic integral of the fiist kind, Vo^JWa/ V.R (x), 
and the formulae (87) furnish the solution of the generalized prob- 
lem of parameter representation. 

82. Curves of deficiency one. An algebraic plane curve C n of degree 
n cannot have more than (n 1) (n 2)/2 double points without 
degenerating into seveial distinct curves. If the curve C n is not 
degenerate and has d double points, the difference 

***~ <T d 

is called the deficiency of that curve. Curves of deficiency zero are 
called unicursal curves , the cooidmates of a point of such a carve 
can be expressed as rational functions of a parameter The next 
simplest curves are those of deficiency one; a cuive of deficiency 
one has (n !)(% 2)/2 1 = n(n 3)/2 double points. 

The coordinates of a point of a curve of deficiency one can be 
expressed as elliptic functions of a parameter. 

In order to prove this theorem, let us consider the adjoint curves 
of the (n 2)th order, that is, the curves C n _ 2 which pass through 
the n(n 3)/2 double points of C n . Since (n 2) (n + 1)/2 points 
are necessary to determine a curve of the (n 2)th degree, the 
adjoint curves C R _ 2 depend still upon 






arbitrary parameters If we also require that these curves pass 
through n 3 other simple points taken at pleasure on C n) we obtain 
a system of adjoint cuives which have, in common with C n , the 
n(n 3)/2 double points of C n and n 3 of its simple points Let 
F(x 9 y)= be the equation of C n , and let 



be the equation of the system of curves C r w _ 2J where X and p are arbi- 
trary parameters. Any curve of this system meets C n m only three 
variable points, for each double point counts as two simple points, 

and we have 

n ( n - 3) 4- n 3 = n (n - 2) - 3. 
Let us now put 



192 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 82 

when the point (a, y) desciibes the curve C n , the point (x\ ?/') de- 
scubes an algebraic cuive C' whose equation would be obtained by 
the elimination of a: and y between the equations (88) and F(x, y) = 
The two cuives C 1 and C n eouesponcl to each other point for point 
by means of a birational transformation This means that, con- 
versely, the cooidinates (it, y) of a point of C n can be expiessed 
lationally in teims of the cooidinates (&', y 1 ) of the corresponding 
point of C" To prove this we need only show that to a point (# f , ?/') 
of C 1 there coriesponds only one point of C n , or that the equations 
(88), togethei with F(x, y) = 0, have only a single system of solu- 
tions foi a and #, which vaiy with x 1 and y\ 

Suppose that to a point of C' there coriespond actually two points 
(a, &), (#', ') of C n which are not among the points taken as the 
basis of the system of curves C n ^ 2 . Then we should have 



a, J) fja, 5) 

and all the curves of the system which pass through the point (a, V) 
would also pass thiough the point (a\ &') The curves of the system 
which pass thiough these two points would still depend linearly 
upon a variable parameter and would meet the curve C n in a single 
variable point The coordinates of this last point of intei section 
with C n would then be lational functions of a variable parameter, 
and the curve C n would be umcursal. But this is impossible, since 
it has only n(n 3)/2 double points Hence to a point (x', y') of C' 
corresponds only one point of C n? and the coordinates of this point 
are, by the theory of elimination, rational functions of x r and y 1 

(89) x = <>', ?/'), y = <i> 2 (x', y') 

In order to obtain the degree of the curve C 1 , let us try to find 
the number of points common to this curve and any straight line 
ax 1 + ly* + o = This amounts to finding the number of points 
common to the curve C n and the curve 



since to a point of C' corresponds a single point of C n , and conversely. 
Now there are only three points of intersection which vary with a, b, c. 
The curve C 1 is therefore of the third degree To sum up, the coor- 
dinates of a point of the curve C n can be expressed rationally in 
terms of the coordinates of a point of a plane cubic , and since the 
coordinates of a point of a cubic are elliptic functions of a parameter, 
the same thing must be true of the coordinates of a point of C n 



HI, Exs ] EXERCISES 193 

It results also from the demonstration, and from what has been 
seen above for cubics, that the representation can be made in such a 
way that to a point (* 7 y) of C n corresponds only one value of it in 
a paiallelogram of periods 

Let x = ^(M), y = ^(w) be the expressions for x and y denved 
above , then every Abehaii integial w = fH (#, ?/) dx associated with 
the curve C n (I, 103, 2d ed , 108, 1st ed.) is reduced by this 
change of vanables to the integral of an elliptic function , hence this 
integial w can be expressed m teims of the transcendental^ p, a- 
of the theory of elliptic functions The mtioduetion of these tian- 
scendentals in analysis has doubled the scope of the integral calculus 

Example Bicircular quwtics A cuive of the fourth degree \\ ith two double 
points is of deficiency one If the double points aie the circulai points at in- 
finity, the curve C 4 is called a bicircular quartic If we take foi the ongm a 
point of the curve, we can take foi the adjoint cuives O n -z cncles passing 
thioughtkeorigm 



In order to have a cubic corresponding point for point to the quartic C 4 , vi e 
need only follow the general method and put x' /(z 2 H- y 2 ), y' = y/(x 2 + y 2 ) 
We have, conversely, x = ir'/^' 2 -f 2/' 2 ), y y'Jtf* + y' 2 ) These formulae define 
an inversion with respect to a cucle of unit radius described with the oiigm 
as centei To obtain the equation of the cubic Cg, it \vill sumee to replace x 
and y in the equation of C 4 by the preceding values Suppose, foi example, 
that the equation of the quartic C 4 is (x 2 + y 2 ) 2 ay = 0, the cubic Cj will 
have for its equation ay'ty* + ' 2 ) 1=0 

Note When a plane curve C n has singular points of a higher order, it is of 
deficiency one, provided that all its singular points are equivalent to n(n 3)/2 
ordinary double points For example, a curve of the fourth degree having a 
single double point at which two branches of the cuive are tangent to each 
other without having any other singularity is of deficiency one ; to verify this 
it suffices to cut the quartic by a system of comes tangent to the two branches 
of the quartic at the double point and passing through another point of the 
quartic. The curve y^ = 12 (jc), where E(x) is a polynomial of the fourth degree 
pume to its denvative, has a smgulanty of this kind at the point at infinity. 
It is reduced to a cubic by the following birational transformation 



x = a;', y = 
from which at is easy to obtain the formulae (87) 

EXERCISES 

1 Prove that an integral doubly periodic function is a constant by means 
of the development 



(The condition f(z + &0 =/() requires that we have A n = If n & ) 



194 SINGLE-VALUED ANALYTIC FUNCTIONS [m,Exs 

2 If fit is not a multiple of TT, we have the formula 



a mr/ 



(Change z to z + a in the expansion for ctn z, then integrate between the 
limits and z ) 

3 Deduce from the preceding result the new infinite products 



a) / 
a \ 



cos a \ Sa-ffly-t-JL 2a (2n ] 

smo: sing 
sin a 

cos z cos a 
1 cos a 

Transform these new products into products of pnmaiy functions or into 
products that no longer contain exponential factors, such as 



-**\. Pi 

97rV L 



T*/\ 97TV f 

4. Derive the relations 
tanz = 2z| : h- 



Establish analogous relations for 

1 1 

, .^ 

sin z sin a cos z cos a 
5. Establish the relation 

^i-g!. 



6 Decompose the functions 

1 

P"(M) ' 
into simple elements. 

7. If flT 2 = 0, we have 

p (CM ; 0, 8 ) = op (u ; 0, gr 8 ), tf( m , 0, f/ 8 ) = p'( w , 0, ^ 8 ), 

where a is one of the cube roots of unity. From this deduce the decomposition 
'K)- p'()] into simple elements when g z = 



Ill, Exs ] EXERCISES 195 

8 Given the integrals 

/ax + b f ax 2 + 6 
-=dx+ I , (%c, 
(x-l)Vx 3 -! ^ J Vl + x 4 ' 

/dx r ax 2 + 6 

x 3Vx3-x* ^ V(1 JC 3 )(1 ^x 2 )^' 



it is required to express the variable x and each one of these integrals in terms 
of the transcendentals p, f, a- 

9 Establish Heimite's decomposition formula ( 73) by equating to zero 
the sum of the residues of the function F(z)[(x, z) (Z Q z)l m a paral- 
lelogiain of penods, where F(x) is an elliptic function and where a, x are 
considered as constants 

10 Deduce fiom the formula (00) the relation 77 = 0'"(0)/12 0'(0). 
(It should be noticed that the series f 01 <r (u) does not contain any terms 
in w 8 ) 

11* Expiess the coordinates x and y of one of the following curves as 
elliptic functions of a parameter 

y=^L[(-a) (x-6) <s-c)p, y = ^[(as-a) (x-5)] 2 , 
y* = ^. (sc a) 2 (z 6) 8 (a; c) 3 , 1^ = ^ (x a) 2 (x 6) 3 , 

y*=ul(x-a)(-6), 

2/ 6 =^L (x-a) 8 (x-6) 4 (-c) 5 , 3^ = ^. (x a) 8 (x - 6) 4 , 
2/6=-d. (x a) 8 (aj &) 5 , y 6 = ^d (x a) 4 (a: 6) 5 , 

- a) (x - &) ( c)] = 0, 



The variable parameter is equal, except for a constant, to the integral /(1/y) <Zx. 
[BRIOT ET BOCQUET, Th&orie desfoncltons doublement 
pfriodzques, 2d ed , pp 388-412.] 



CHAPTER IV 
ANALYTIC EXTENSION 

I DEFINITION OF AN ANALYTIC FUNCTION BY MEANS 
OF ONE OF ITS ELEMENTS 

83. Introduction to analytic extension. Let f(z) be an analytic func- 
tion in a connected poition A of the plane, bounded by one or more 
cuives, closed 01 not, wheie the word curve is to be undei stood in 
the usual elementaiy sense as heretofore 

If -we know the value of the function f(z) and the values of all 
its successive derivatives at a definite point a of the region A, we 
can deduce from them the value of the function at any othei point b 
of the same region To piove this, ]oin the points a and I by a path L 
lying entnely in the region A , for example, by a bioken line 01 by 
any foim of cuive whatever. Let 8 be the lower limit of the dis- 
tance from any point of the path L to any point of the boundary of 
the region A, so that a ciicle with the radius 8 and with its centei at 
any point of L will lie entirely in that region By hypothesis we 
know the value of the function f(a) and the values of its successive 
derivatives f r (a), /"(&), , for & = a We can therefore write the 
power series which represents the function /(#) in the neighborhood 
of the point a . 



(1) 



The radius of convergence of this series is at least equal to 8, but 
it may be greater than 8 If the point I is situated in the circle of 
convergence C of the preceding senes, it will suffice to replace by 
b in order to have/(Z>) Suppose that the point b lies outside the circle 
C , and let <Xi be the point where the path L leaves C * (Fig. 30) 
Let us take on this path a point 1 within C and near a v so that the 

* Since the value of f(z) at the point 6 does not depend on the path so long as it 
does not leave the region A, we may suppose that the path cuts the ciicle GQ in only 
one point, as in the figure, and the successive circles Cj, C" 2 , in at most two points 
This amounts to taking for ^ the last point of intersection of L and <? , and similarly 
for the others 

196 



IV, 83] ELEMENTS OF AN ANALYTIC FUNCTION 



197 



distance between the two points ^ and ^ shall be less than S/2 The 
series (1) and those obtained from it "by successive differentiations 
enable us to calculate the values of the function f(z) and of all its 
derivatives, /(*,), f(*J, . . , /<) ( gj ), . , f or * = * r The coefficients 
of the series which represents the function /(*) in the neighborhood 
of the point z l are therefore determined if we know the coefficients of 
the fiist series (1), and we have m the neighborhood of the point ^ 



The radius of the circle of convergence C^ of this series is at least 
equal to 8; this circle contains, then, the point <K : within it, and 
there is also a part of it out- 
side of the circle C . If the 
point I is in this new circle 
C v it will suffice to put z = # 
in the series (2) in order to 
have the value of /(). Sup- 
pose that the point I is again 
outside of C v and let a 2 be 
the point where the path z^b 
leaves the circle. Let us take 
on the path L a point # 2 
within C l and such that the 

distance between the two points # 2 and a 2 shall be less than 5/2 
The series (2) and those which we obtain from it by successive dif- 
ferentiations will enable us to calculate the values of f(z) and its 

derivatives /(*,), /(^ f'(*d> * * * at tiie P omt 
form a new series, 




(3) 



O + 



which represents the function f(&) in a new circle C 2 with a radius 
greater than or equal to 8. If the point b is in this circle C 2 , we shall 
replace # by b in the preceding equality (3); if not, we shall continue 
to apply the same process. At the end of a finite number of such 
operations we shall finally have a circle containing the point b within 
it (in the case of the figure, b is in the interior of C^) , for we can 
always choose the points v 3 , # 8 , - in such a way that the dis- 
tance between any two consecutive points shall be greater than S/2. 
On the other hand, let 5 be the length of the path . The length of 



198 ANALYTIC EXTENSION [IV, 83 

the broken line az^ 2 ^ p -^ p is always less than S 9 hence we have 
^s/2 + \z p b\<S Let p be an integer such that (p/2 + 1) 8 > S 
The piecedmg inequality shows that aftei p opeiations, at most, 
we shall come upon a point z p of the path L whose distance from 
the point I will be less than 8, the point b will be in the mtenor 
of the circle of convergence C f of the power series which represents 
the function /() in the neighboihood of the point p) and it will 
suffice to replace # by I in this series in oider to have/(Z>) In the 
same way all the derivatives /'(), /"(&), * can be calculated 

The above reasoning proves that it is possible, at least theoretically, 
to calculate the value of a function analytic in a region A, and of 
all its derivatives at any point of that region, provided we know 
the sequence of values, 



of the function and of its successive derivatives at a given point a of 
that region It follows that any function analytic in a region A is 
completely determined in the whole of that region if it is known in 
a region, however small, surrounding any point a taken in A, or 
even if it is known at all points of an arc of a curve, however shoit, 
ending at the point a For if the function f(z) is determined at 
every point on the whole length of an arc of a curve, the same must 
be true of its derivative /'(), since the value f(zj at any point of 
that arc is equal to the limit of the quotient [/(V,) /(X)]/^ x ) 
when the point z 2 appi caches z 1 along the arc considered , the deriv- 
ative/'^) being known, we deduce from it in the same way /"(), 
and from that we deduce /"'(*), All the successive derivatives 
of the function /() will then be detei mined for # = a. We shall say 
for brevity that the knowledge of the numerical values of all the 
terms of the sequence (4) determines an element of the function 
/(). The result reached can now be stated in the following man- 
ner : A function analytic in a region A is completely determined if 
we know any one of its elements We can say further that two func- 
tions analytic in the same region cannot have a common element 
without being identical. 

We have supposed for definiteness that the function considered, 
/(), was analytic in the whole region; but the reasoning can be 
extended to any function analytic in the region except at certain 
singular points, provided the path L, followed by the variable in 
going from a to 5, does not pass through any singular point of the 
function. It suffices for this to break up the path into several arcs, 



IV, 84] ELEMENTS OF AN ANALYTIC FUNCTION 199 

as we have already done ( 31), so that each one can be inclosed 
in a closed boundary inside of which the branch of the function /( ^) 
considered shall be analytic The knowledge of the initial element 
and of the path described by the variable suffices, at least theoieti- 
cally, to find the final element, that is, the numencal values of all the 
terms of the analogous sequence 



84. New definition of analytic functions. Up to the present we have 
studied analytic functions which were defined by expressions which 
give their values for all values of the variable in the field in which 
they were studied. "We now know, fiom what precedes, that it is 
possible to define an analytic function for any value of the variable 
as soon as we know a single element of the function ; but in order to 
present the theory satisfactorily from this new point of view, we must 
add to the definition of analytic functions accoiding to Cauchy a new 
convention, which seems to be woith stating in considerable detail. 

Let/ x (), f 2 (z) be two functions analytic respectively in the two 
regions A I} A having one and only one part 
A* in common (Fig. 31) If in the com- 
mon part A r we have / 2 (s) = / 1 (s), which 
will be the case if these two functions have 
a single common element in this region, we 
shall regard f^z) and / 2 (^) as forming a 
single function F(z), analytic in the region 
A^AQ, by means of the following equalities: 

*(*) =/i(*) m A v and F (*)=/2(*) in A r 

We shall also say that f z (&) is the analytic extension into the region 

A 2 A' of the analytic f unction ffa), which is supposed to be defined 
only in the region A^ It is clear that the analytic extension of / t () 
into the region of A Z exterior to A l is possible in only one way.* 

*In order to show that the preceding convention is distinct from the definition of 
functions analytic m general, it suffices to notice that it leads at once to the following 
consequence IJ a f unction f(z) i& analytic in a region A, every other analytic func- 
tion f fa), under these conventions, which coincides withf(z) in a part of the region A 
is identical withf(z) in A Now let us consider a function F(z) defined for all values 
of the complex variable z in the following manner 




F(z) sin , if z 5* -> F\^J = 

However odd this sort of convention may appear, it has nothing in it contra- 
dictory to the previous definition of functions m general analytic. The function 
thus defined would he analytic for all values of z except for z = ?r/2, which would 



200 ANALYTIC EXTENSION [IT, 84 

Let us now consider an infinite sequence of numbers, real or 
imaginary, 

(6) ; a v a z > 3 w ? - 9 

subject to the single condition that the series 

CO o + V + ^ +' + a *" + * ' ' 

converges for some value of 2 different from zero (We take 2 = 
for the initial value of the variable, which does not in any way 
restrict the generality.) The series (7) has, then, by hypothesis, a 
cucle of convergence C Q whose radius R is not zero If R is infinite, 
the series is convergent for every value of # and represents an inte- 
gral function of the variable. If the radius R has a finite value dif- 
ferent from zero, the sum of the series (7) is an analytic function 
f(z) in the interior of the circle C But since we know only the 
sequence of coefficients (6), we cannot say anything a priori regard- 
ing the natuie of the function outside of the cucle (7 We do not 
know whether or not it is possible to add to the circle C f an adjoin- 
ing region forming with the circle a connected region A such that 
there exists a function analytic in A and coinciding with/(^) in the 
interior of C , The method of the preceding paragiaph enables us to 
determine whether this is the case or not Let us take in the circle C 
a point a different from the origin By means of the series (7), 
and the senes obtained from it by term-hy-term differentiation we 
can calculate the element of the f unction /() which eoi responds to 
the point a, and consequently we can form the power series 



(8) 

which represents the function/^) in the neighborhood of the point a 
This series is certainly convergent m a circle about a as center with 
a radius R a\ ( 8), but it may be convergent in a larger circle 
whose radius cannot exceed R + \a\. Por if it were convergent in 

be a singular point of a particular nature But the properties of this function F(z) 
would be in contradiction to the convention which, we have just adopted, since the 
two functions F(z) and sm z would be identical for all the values of z except for 
z = ir/2, which would be a singular point for only one of the two functions 

Weierstrass, in Germany, and Meray, in France, developed the theory of analytic 
functions by starting only with the properties of power series, their investigations 
are also entirely independent Meray's theory is presented m his large treatise, 
Lemons nouveltes $ur F Analyse tttfimtesimale It is shown in the text how we can 
define an analytic function step by step, knowing one of its elements but always 
supposing known the theorems of Cauchy on analytic functions 



IV, 84] ELEMENTS OF AN ANALYTIC FUNCTION 201 

a circle of radius R -f \a\ + S, the senes (7) would be con vei gent in 
a cucle of ladius R -f- 8 about the origin as center, contiary to the 
hypothesis. Let us suppose first that the radius of the circle of con- 
vergence of the series (8) is always equal to R ||, wheiever the 
point a may be taken in the circle C Then there exists no means 
of extending the function /fc) analytically outside of the ciicle, at 
least if we make use of power senes only We can say that there 
does not exist any function F(z) analytic in a region A of the plane 
gieater than and containing the cncle C Q and coinciding with f(z) 
in the circle C , for the method of analytic extension would enable 
us to determine the value of that function at a point exterior to the 
circle C , as we have ]ust seen The cucle C is then said to be a 
natural "boundary foi the function f(?) Furthei on we shall see 
some examples of this 

Suppose, in the second place, that with a suitably chosen point 
a in the cucle C the cucle of conveigence C^ of the series (8) has a 
radius greater than R \ a \ G 

This circle C l has a part 
exterior to C (Fig 32), and 
the sum of the senes (8) is 
an analytic function / x () in 
the circle C l In the interior 
of the circle y with the center 
a, which is tangent to the 
circle C internally, we have 
/!()=/(*) (8); hence this 
equality must subsist in the 
whole of the region common 
to the two circles C , C l The 
senes (8) gives us the analytic extension of the function /() into 
the portion of the circle C l exterioi to the cucle C Let a* be a new 
point taken in this region , by proceeding in the same way we shall 
form a new power senes in powers of & a', whi6h will be con- 
vergent m a circle C 2 If the cucle C 2 is not entirely within C 1? the 
new senes will give the extension of /() in a more extended region, 
and so on in the same way. We see, then, how it is possible to 
extend, step by step, the region of existence of the function /(), 
which at first was defined only in the interior of the ciicle C . 

It is clear that the preceding process can be carried out in an in- 
finite number of ways In order to keep in mind how the extension 
was obtained, we must define precisely the path followed by the 




202 ANALYTIC EXTENSION [IV, 84 

variable Let us suppose that we can obtain the analytic extension 
of the function denned by the series (7) along a path L, as we have 
just explained Each point x of the path L is the center of a circle of 
conveigence of radius r in the mteiior of which the function is lep- 
resented by a con vei gent senes arranged in powers of z x The 
radius r of this circle varies continuously with x For let x and x 1 be 
two neighboung points of the path L } and r and r' the corresponding 
radii If x r is near enough to x to satisfy the inequality \x' x \ < r, 
the radius r' will lie between r | x 1 x \ and r + \x' x |, as we have 
seen above Hence the difference r f r appi caches zero with \x* x \ 
Now let C' Q be a cucle with the ladius R/2 descubed with the origin 
as center; if a is any point on the circle C' QJ the ladius of conver- 
gence of the senes (8) is at least equal to 7?/2, but it may be greater 
Since this iadms vanes in a continuous manner with the position of 
the point a, it passes thiough a minimum value R/2 + r at a point 
of the circle C' Q We cannot have r > 0, for if r weie actually posi- 
tive, theie would exist a function F(z) analytic in the circle of radius 
22 _j_ r about the ongin as center and coinciding with /(&) in the 
interior of C For a value of z whose absolute value lies between R 
and R + r, F(z) would be equal to the sum of any one of the series 
(8), where a is a point on CJ such that [ z a \ < R/2 + r Accoiding 
to Cauchy's theoiem, F(z) would be equal to the sum of a power 
series convergent in the circle of radius R + r, and this series would 
be identical with the senes (7), which is impossible. 

There is, therefore, on the circumference of C' at least one point a 
such that the circle of convergence of the series (8) has R/2 for its 
iadms, and this circle is tangent internally to the cncle C Q at a point 
a where the radius Oa meets that circle The point <z is a singular 
point of /(#) on the circle C In a cucle c with the point a for 
center, however small the radius may be taken, there cannot exist 
an analytic function which is identical with/(#) in the part common 
to the two circles C Q and c. It is also clear that the circle of conver- 
gence of the sferies (8) having any point of the radius Oa foi center 
is tangent internally to the circle C at the point a * 

* If all the coefficients a* of the series (7) are real and positive, the point z-Ris 
necessarily a singular point on C7 In fact, if it were not, the power senes 



which represents/ (2) in the neighborhood of the point z = R/2, would have a radius 
of convergence greater than jR/2 The same would he true a fortiori of the series 



IV, 84] ELEMENTS OF AN ANALYTIC FUNCTION 203 

Let us consider now a path. L starting at the origin and ending at 
any point Z outside of the circle C 0? and let us imagine a moving 
point to describe this path, moving always in the same sense from 
to Z Let a^ be the point where the moving point leaves the circle ; 
if this point 0J were a singular point, it would be impossible to con- 
tinue on the path L beyond this point We shall suppose that it is 
not a singular point , we can then form a power series arranged in 
powers of or x and convergent in a cncle C x with the center a^ 
whose sum coincides with f(z) m the part common to the two cir- 
cles C and C\ To calculate /(^), /'(a? a ), we could employ, for 
example, an intermediate point on the radius Oa % The sum of the 
second series would furnish us with the analytic extension of f(z) 
along the path L from a v so long as the moving point does not leave 
the circle C t In particular, if all the path starting from ^ lies in 
the interior of C 19 that series will give the value of the function at the 
point Z If the path leaves the circle C 1 at the point or 2 , we shall 
form, similarly, a new power series convergent in a circle C 2 with 
the center # 2 , and so on* We shall suppose first that after a finite 
numbei of operations we arrive at a circle C p with the center a p , con- 
taining all the portion of the path L which follows a p9 and in partic- 
ular the point Z It will suffice to replace z by Z in the last series 
used and in those which we have obtained fiom it by term-by-term 
differentiation in order to find the values of /(Z), f(Z}, f'(Z\ -, 
with which we arrive at the point Z, that is, the final element of the 
function. 

It is clear that we arrive at any point of the path L with com- 
pletely determined values for the function and all its derivatives 
Let us note also that we could replace the circles C , C^ C 29 * -, O p 
by a sequence of circles similarly defined, having any points z l9 # 2 , 
, & q of the path L as centers, provided that the circle with the 
center ^ contains the portion of the path L included between z v and 
z i+i We can also modify the path L, keeping the same extremities, 
without changing the final values of /(#), /'(#), /"(), 5 for the 



whatever the angle a may be, for we have evidently 



since all the coefficients a are positive The minimum of the radius of convergence 
of the series (8) , when a describes the circle Cj, would then be greater than JR/2 



204 



ANALYTIC EXTENSION 



[IV, 84 




FIG 33 



circles C , C v - , C f cover a portion of the plane forming a kind of 
strip in -which the path L lies, and we can replace the path L by any 
other path L 1 going from 2 = to the point Z and situated in that 

strip Let us suppose, for 
definiteness, that we have to 
make use of three consecutive 
circles C , C 1? C 2 (Fig 33) 
Let L r be a new path lying 
in the strip formed by these 
three circles, and let us join 
the two points m and n If we 
go from to m first by the 
path Oa^m, then by the path 
Onm, it is clear that we arrive 
at m with the same element, since we have an analytic function in 
the region formed by (7 and C l Similarly, if we go from m to Z 
by the path ma^Z or by the path mnZ, we arrive in each case at 
the point Z with the same element The path L is therefore equiv- 
alent to the path OnmnZ, that is, to the path V The method of 
proof is the same, whatever may be the number of the successive 
cucles. In particular, we can always replace a path of any form 
whatever by a bioken line* 

85. Singular points. If we proceed as we have just explained, it 
may happen that we cannot find a circle containing all that part of 
the path L which remains to be descubed, however far we continue 
the process This will be the case when the point a p is a singular point 
on the ciicle C p _ ly for the process will be checked just at that point 
If the process can be continued forevei, without arriving at a circle 
inclosing all that pait of the path L which lemains to be described, 
the points tf p _ 1? a p , a p+ i, approach a limit point X of the path Z, 
which may be either the point Z itself or a point lying between 
and Z. The point A is again a singular ijoint, and it is impossible 
to push the analytic extension of the unction /() along the path L 
beyond the point X. But if X is different from Z, it does not follow 
that the point Z is itself a singular point, and that we cannot go 
from O to Z by some other path. Let us consider, for example, either 
of the two functions Vl-hs and Log (1 + 2) , we could not go from 

* The reasoning requires a little more attention when the path L has double points, 
since then the strip formed hy the successive circles <7 , Ci, <7 2 , may return and 
cover part of itself But there is no essential difficulty 



IV, 85] ELEMENTS OF AN ANALYTIC FUNCTION 205 

the origin to the point = 2 along the axis of reals, since we could 
not pass through the singular point 2 = 1 But if we cause the van- 
able z to describe a path not going through this point, it is clear that 
we shall arrive at the point z = 2 after a finite number of steps, 
for all the successive circles will pass through the point z = 1 
It should be noticed that the preceding definition of singular points 
depends upon the path followed by the variable , a point X may be 
a singular point foi a certain path, and may not for some other, if 
the function has several distinct branches 

When two paths L 19 L{, going from the origin to Z, lead to dif- 
ferent elements at Z, there exists at least one singulai point in the 
interior of the legion which would be swept out by one of the paths, 
L 19 for example, if we were to deform it in a continuous manner so 
as to bring it into coincidence with L(, retaining always the same 
extremities duung the change Let us sup- 
pose, as is always permissible, that the two 
paths L 19 L{ are broken lines composed of the 
same number of segments Oa^q lZ and 
Oa&t 1{Z (Fig 34) Let a# b 2 , c 2 , . , 1 2 
be the middle points of the segments a^a'^ 
bib'i) c i c iy * ? W? the path L^ formed by the 
broken line a 2 # 2 c 2 l^Z cannot be equiva- 
lent at the same time to the two paths L 19 L{ 
if it does not contain a singular point If the 
path Z 2 does contain a singular point, the 
theorem is established If the two paths L^ 
and L^ are not equivalent, we can deduce from 
them a new path 8 lying between L^ and L 2 
by the same process Continuing in this way, we shall either reach 
a path L p containing a singular point or we shall have an infinite 
sequence of paths L 19 2 , 8 , . These paths will approach a limit- 
ing path A, for the points a l9 a a , c& 8 , approach a limit point lying 
between a x and a(, , and similarly foi the others This limiting 
path A must necessanly contain a singular point, since we can 
draw two paths as near as we please to A, one on each side of 
it, and leading to different elements for the function at Z, This 
could not be true if A did not contain any singular points, since 
the paths sufficiently close to A must lead to the same elements 
at Z as does A 

The preceding definition of singular points is purely negative 
and does not tell us anything about the nature of the function in 




206 ANALYTIC EXTENSION [IV, 85 

the neighboihood. ~No hypothesis on these singular points or on 
their distubution in the plane can be discarded a pnon without 
danger of leading to some contradiction A study of the analytic 
extension is required to determine all the possible cases.* 

86. General problem. From what precedes, it follows that an analytic 
function is virtually detei mined when we know one of its elements, 
that is, when we know a sequence of coefficients a Q , a v a a , , a n , 
such that the series 

a Q + a^(x a)+ + a n (x <x) n H 

has a radius of conveigence different from zero. These coefficients 
being known, we are led to consider the following general problem 
To find the value of the function at any point p of the plane when the 
variable is made to describe a definitely chosen path from the point a 
to the point f$ We can also consider the problem of determining 
a prioii the singular points of the analytic function, it is also 
clear that the two problems are closely related to each other The 
method of analytic extension itself furnishes a solution of these two 
problems, at least theoretically, but it is piaeticable only in very 
particular cases For example, as nothing indicates a priori the 
number of intermediate series which must be employed to go from 
the point a to the point ft, and since we can calculate the sum of 
each of these series with only a certain degree of approximation, it 
appears impossible to obtain any idea of the final approximation 
which we shall reach So the investigation of simpler solutions was 
necessaiy, at least in particular cases Only in recent years, how- 
ever, has this problem been the object of thorough investigations, 
which have already led to some impoitant results t 

*Let/(a;) be a function analytic along the whole length of the segment ab of the 
real axis. In the neighboihood of any point <x of this segment the function can be 
represented by a power series whose radius of convergence J2(ar) is not zero This 
radius R, being a continuous function of or, has a positive minimum r Let p be a 
positive number less than r, and E the region of the plane swept out by a circle with, 
the radius p when its center describes the segment ab The function/ () is analytic 
In the region E and on its boundary , let M be an upper bound f 01 its absolute value , 
from the general formulae (14) ( 33) it follows that at any point ar of ab we have the 
inequality 

l/l<7-r 

(Of, I, 19T, 2d ed , 191, 1st ed ) 

fFor everything regarding this matter we refer the reader to Hadamard's excel- 
lent work, La srze de Taylor et son prolongement analytique (Naud, 1901). It con- 
tains a very complete bibliography. 



IV, 86] ELEMENTS OF AN ANALYTIC FUNCTION 207 

The fact that these researches are so recent must not be attubuted 
entirely to the difficulty of the question, however great it may be 
The functions which have actually been studied successively by 
mathematicians have not been chosen by them aibitrarily ; rather, 
the study of these functions was forced upon them by the very nature 
of the pioblems which they encounteied Now, aside from a small 
numbei of transcendentals, all these functions, after the explicit 
elementary functions, aie defined either as the roots of equations 
which do not admit a foimal solution 01 as integrals of algebiaic 
differential equations It is clear, then, that the study of implicit 
functions and of functions defined by differential equations must 
logically have preceded the study of the geneial problem of which 
these two problems are essentially only very paiticular cases. 

It is easy to show how the study of algebraic differential equa- 
tions leads to the theory of analytic extension Let us consider, for 
concreteness, two power senes 2/(r), z(x) } arranged according to pos- 
itive powers of x and convergent in a cucle C of radius R descubed 
about the point x = as center On the other hand, let F(x y y, y r , y") 
, yM, z, *', , *to>) be a polynomial in x, y, y\ - , y^\ z, z',-- , z (< *\ 
Let us suppose that we replace y and & in this polynomial by the 
preceding senes, y\ y", , y^ by the successive derivatives of the 
series y(x), and z' } z", , z^ by the derivatives of the series #(#), 
the result is again a power series convergent an the circle C. If all 
the coefficients of that series are zero, the analytic functions y(x) 
and (x) satisfy, in the circle C, the relation 



(9) F(x,y,y', , y<*\ z, z', 

We are now going to prove that the functions obtained by the analytic 
extension of the series y(x) and z(x) satisfy the same relation in the 
whole of their domain of existence ** Moie precisely, if we cause the 
variable x to describe a path L s tax ting at the origin and proceeding 
fiom the circle C to reach any point a of the plane, and if it is pos- 
sible to continue the analytic extension of the two series y(x) and 
' (cr) along the whole length of this path without meeting any singular 
point, the power series Y(x a) and Z(x a) with which we arrive 
at the point a represent, in the neighborhood of that point, two ana- 
lytic functions which satisfy the relation (9) For let x l be a point 
of the path L within the circle C and near the point where the path L 
leaves the circle C With the point a; 1 as center we can describe a 
circle C 19 partly exterior to the circle C, and there exist two power 
series y(x x^), z(x x^) that are convergent in the circle C l and 



208 ANALYTIC EXTENSION [IV, 86 

whose values are identical with, the values of the two series y (x) and 
z(x) in the part common to the two circles C, C : Substituting for y 
and s in Fthe two corresponding series, the result obtained is a power 
series P(x x^ eon vei gent in the circle C l Now in the part common 
to the two circles C, C 1 we have P(x - c a )= 0, the series P(x - xj 
has therefore all its coefficients zero, and the two new series y(x xj 
and z (x #,) satisfy the relation (9) in the circle C l Continuing 
in this way, we see that the relation never ceases to be satisfied 
by the analytic extension of the two senes y(x) and #(#), whatever 
the path followed by the variable may be, the proposition is thus 
demonstrated. 

The study of a function defined by a differential equation is, then, 
essentially only a particular case of the general problem of analytic 
extension But, on the other hand, it is easy to see how the knowledge 
of a particular relation between the analytic function and some of 
its derivatives may in certain cases facilitate the solution of the 
problem We shall have to return to this point in the study of 
differential equations. 

n NATURAL BOUNDARIES. CUTS 

The study of modular elliptic functions furnished Hermite the 
first example of an analytic function defined only in a portion of 
the plane We shall point out a very simple method of obtaining 
analytic functions having any curve whatever of the plane for a 
natural boundary (see 84), under certain hypotheses of a very 
general character concerning the curve 

87. Singular lines. Natural boundaries. We shall first demonstrate 
a preliminary proposition * 

Let (j&j, a 2 , , a n , and c 1? c 2 , , c n , * be two sequences of 
any kind of terms, the second of which is such that $c v is absolutely 
convergent and has all its terms different from zero Let C be a 
circle with the center Z Q , containing none of the points & t in its interior 
and passing through a single one of these points , then the series 



s 



*POIXCARI, Acta Societatis Fenmcse, Vol XIII, 1881, GOUESAT, Bulletin ties 
sciences mathematiqiLes, 2d senes, Vol XI, p 109, and Vol XVII, p 247 



IV, 87] NATURAL BOUNDARIES CUTS 209 

represents an analytic function in the circle C which can be devel- 
oped in a sei les of powers of z # The circle of convergence of this 
set les is precisely the circle C 

We can cleaily suppose that Z Q = 0, for if we change & to Z Q &', 
a v is replaced by a v # , and c v does not change We shall also sup- 
pose that we have | aj = R, where R denotes the ladms of the circle C, 
and | Ofc| > R for i > 1 In the circle C the general teim c v /(a v z) can 
be developed in a power series, and that series has (\e v \/R)/(L z/R) 
for a dominant function, as is easily verified By a general theorem 
demonstrated above ( 9), the series S|<3 V | being convergent, the func- 
tion F(z) can be developed in a power series in the circle C, and that 
series can be obtained by adding term by term the power series which 
represent the different terms We have, then, in the circle C 

(10') F(z) = A Q + A l z + A z z* + + A n z + , ^JgJ^. 

v l^v 

+ CO 
Let us choose an integer p such that V|c^| shall be smaller than 

v=p+l 

|cJ/2, which is always possible, since c 1 is not zero and since the 
series S|c,,| is convergent Having chosen the integer p in this 
way, we can write F(z) = F^z) + F z (z\ where we have set 



a v z 

v 



v -&-. 

*-/ a v z 

v 



F^(z) is a rational function which has only poles exterior to the 
circle C ; it is therefore developable in a power series in a circle C f 
with a radius R 1 > R As for F 2 (z), we have 

(11) JP a ()=5 + J5 1 *+ +-^+ , 

where 

7? C l _i C P+1 i gp + 2 _|._ 

^ - + 



We can write this coefficient again in the form 



but we have, by hypothesis, |a 1 / ft j< 1, and the absolute value of 
the sum of the series 



210 ANALYTIC EXTENSION [IV, 87 

is less than [flJ/2, by the method of choosing the integer p. The 
absolute value of the coefficient B n is theref 01 e between (cJ/2 J? n+1 and 
3|c |/2 R n+l in magnitude, and the absolute value of the general teini 
of the series (11) lies between (|q|/2 22) \v/R\* and (3|0J/2)|*/K|, 
that series is theref oie divergent if \\ > R By adding to the senes 
F 2 (z), convergent m the circle with the radius R, a series JF^*), con- 
vergent in a circle of radius R 1 > R, it is clear that the sum F(z) has 
the circle C with the radius R for its circle of convergence ; this 
proves the proposition which was stated 

Let now L be a curve, closed or not, having at each point a definite 
radius of curvatuie. The series 50,, being absolutely convergent, let 
us suppose that the points of the sequence a v & 2 , , a t , are all 
on the curve L and are distributed on it in such a way that on a 
finite arc of this curve there are always an infinite number of points 
of that sequence The senes 

(12) *<*)-!" 



, z 



is convergent for every point not belonging to the curve Z, and 
represents an analytic function in the neighborhood of that point 
To prove this it would suffice to repeat the first part of the preced- 
ing pi oof, taking for the cncle C any circle with the center # and 
not containing any of the points a l If the cuive L is not closed, 
and does not have any double points, the series (12) represents an 
analytic function in the whole extent of the plane except for the 
points of the curve L. We cannot conclude from this that the 
curve L is a singular line; we have yet to assure ourselves that 
the analytic extension of F(&) is not possible across any portion 
of L, however small it may be. To prove this it suffices to show that 
the circle of convergence of the power series which represents F(z) 
in the neighborhood of any point 2 not on L can never inclose an 
arc of that curve, however small it may be Suppose that the circle C, 
with the center Z Q) actually incloses an arc aft of the curve L Let us 
take a point ar t on this arc aft, and on the normal to this arc at a % let 
us take a point ' so close to the point a t that the circle C t , described 
about the point z' as center with the radius | r a t [, shall lie entirely 
in the interior of C and not have any point in common with the 
arc aft other than the point a l itself. By the theorem which has ]ust 
been demonstrated, the circle C l is the circle of convergence for the 
power series which represents F(z) in the neighborhood of the point 
z r . But this is in contradiction to the general properties of power 



IV, 88] NATURAL BOUNDARIES CUTS 211 

senes, for that circle of convergence cannot be smaller than the 
circle with the center z 1 which is tangent mteinally to the circle C 

If the curve L is closed, the series (12) represents two distinct 
analytic functions One of these exists only in the interior of the 
curve L } and for it that cuive is a natural boundary, the other 
function, on the contrary, exists only in the region exterior to the 
curve L and has the same curve as a natural boundary Thus the 
curve L is a natural boundary for each of these functions 

Given several curves, L v L^ , L p , closed or not, it will be pos- 
sible to form in this way series of the form (12) having these curves 
for natural boundaries , the sum of these series will have all these 
curves for natural boundaries. 

88. Examples. Let AB be a segment of a straight line, and or, /3 the complex 
quantities representing the extremities A,B All the points 7 = (ma + np)/(m + n) , 
where m and n are two positive integers varying from 1 to + oo, are on the seg- 
ment AB, and on a finite portion of this segment there are always an infinite 
number of points of that kind, since the point 7 divides the segment AB in the 
ratio m/n On the other hand, let C m , n be the geneial term of an absolutely 
convergent double series The double series 



ma + nff 



*<)= 



represents an analytic function having the segment AB for a natural boundary 
We can, in fact, transform this series into a simple series with a single index 
xn an infinite number of ways It is clear that by adding several series of this 
kind it will be possible to form an analytic function having the perimeter of 
any given polygon as a natural boundary 

Another example, m which the curve L is a circle, may be defined as follows 
Let a. be a positive irrational number, and let v be a positive integer Let us put 

a = c 2 17ra , CL V = CL V = B Z I7rva . 

Then all the points a v are distinct and are situated on the circle C of unit radius 
having its center at the origin Moreover, we know that we can find two inte- 
gers m and n such that the difference 27r(nar m) will be less in absolute value 
than a number e, however small c be taken 

There exist, then, powers of a whose angle is as near zero as we wish, and 
consequently on a finite arc of the circumference there will always be an infinite 
number of points a v . Let us next put c v = a v /2 v , the series 



represents, by the general theorem, an analytic function in the circle 
which has the whole circumference of this cncle for a natural boundary 



212 ANALYTIC EXTENSION [IV, 88 

Developing each term in powers of z, we obtain for the development of F(z) the 
power series 






It is easy to prove dnectly that the function represented by this power series 
cannot be extended analytically beyond the circle C , for if we add to it the 
series for 1/(1 z), there results 



2 v ' 21-z 
Changing in this relation s to az, then to #%, , we find the general relation 

(14) F(a n z) = F(z) -I j fr* * -J > 

which shows that the difference 2 n F(a n z) F(z) is a rational function <f> (z) hav- 
ing the n poles of the first order 1, I/a, - , I/a 11 - 1 . 

The result (14) has been established on the supposition that we have \z\ <1 
and | a \ = 1 If the angle of a is commensurable with ?r, the equality (14) shows 
that F(z) is a rational function , to show this it would suffice to take for n an 
integer such that a n = 1 If the angle of a is incommensurable with TT, it is im- 
possible for the function F(z) to be analytic on a finite arc AB of the ciicum- 
ference, however small it may be. For let a-? and #-* be two points on the 
arc AB(n>p) The numbers n and $ having been chosen in this way, let us 
suppose that z is made to approach arf , OPZ will approach a*--?, and the two 
functions F(z) and F(a n z) would approach finite limits if F(z) were analytic 
on the arc AB. Now the relation (14) shows that this is impossible, since the 
function <j> (z) has the pole or P. 

An analogous method is applicable, as Hadamard has shown, to the series 
considered by Weierstrass, 

(15) 

where a is a positive integer > 1 and 6 is a constant whose absolute value is less 
than one This series is convergent if | z \ is not greater than unity, and diver- 
gent if | | is greater than unity The circle C with a unri radius is therefore the 
circle of convergence. The circumference is a natural boundary for the func- 
tion F(z) IPor suppose that there are no singular points of the function on a 
finite arc ap of the circumference If we replace the variable & in F(z) by 
2C 2 * W/C *, where k and h are two positive integers and c a divisor of a, all the 
terms of the series (15) after the term of the rank h are unchanged, and the 
difference F(z) F(z&*-'<*) is a polynomial Neither would the function F(z) 
have any singular points on the arc ojft., which is denved from the arc ocp by a 
rotation through an angle 2 kir/c h around the origin Let us take h large enough 
to make 2 ?r/c* smaller than the arc aft , taking successively k = 1, 2, - * , c*, it 
is clear that the arcs o^, <*a/3 2 , . . cover the circumference completely The 



IV, 89] NATURAL BOUNDABIES CUTS 213 

function F(z) would therefore not have any smgulai points on the circumfer- 
ence, which is absurd ( 84) 

This example presents an interesting peculiarity, the series (15) is absolutely 
and uniformly convergent along the circumference of C It represents, then, a 
continuous function of the angle along this circle * 

89 Singularities of analytical expressions. Every analytical expres- 
sion (such as a series whose different terms are functions of a vari- 
able #, or a definite integral in which that variable appears as a 
parameter) represents, under certain conditions, an analytic function 
in the neighborhood of each of the values of z for which it has a 
meaning If the set of these values of z covers completely a connected 
region A of the plane, the expiession considered represents an 
analytic function of z in that region A , but if the set of these values 
of z forms two or more distinct and separated regions, it may happen 
that the analytical expression considered represents entirely distinct 
functions in these different regions We have already met an exam- 
ple of this in 38 There we saw how we could form a series of 
rational terms, convergent in two curvilinear triangles PQR, P'Q'R* 
(Fig. 16), whose value is equal to a given analytic function f(z) in 
the triangle PQR and to zero in the triangle P'Q'R* By adding two 
such series we shall obtain a series of rational terms whose value is 
equal to f(z) in the triangle PQR and to another analytic function 
<f) (z) in the triangle P'Q'R'. These two functions f(z) and <j> (z) being 

* Fredholm has shown, similarly, that the function represented by the series 



where a is a positive quantity less than one, cannot be extended beyond the circle of 
convergence (Comptes rendus, March 24, 1890) This example leads to a result which 
is worthy of mention On the circle of unit radius the senes is conveigent and the 
value 

F(0) S o[cos (r$0) + 1 sm (n 2 0)] 

is a continuous function of the angle 9 which has an infinite number of derivatives 
This function F(9) cannot, however, be developed in a Taylor's series in any interval, 
however small it may be Suppose that in the interval (#o - a, 0o + a) we actually 
have 



The series on the right represents an analytic function of the complex variable 6 in 
the circle c with the radius a described with the point Q for center To this circle c 
corresponds, by means of the relation z- e fl , a closed region A of the plane of the vari- 
able z containing the arc 7 of the unit circle extending from the point with the angle 
- <* to the point with the angle + <* Tnere would exist, then, in this region A 
an analytic function of z coinciding with the value of the series Sa^ 8 along 7 and also 
m the part of A within the unit circle, this is impossible, since we cannot extend the 
sum of the senes beyond the circle 



214 ANALYTIC EXTENSION [IV, 89 

arbitiary, it is clear that the value of the series in the triangle P'Q'R' 
will in general bear no i elation to the analytic extension of the value 
of that series in the triangle PQR 

The following is anothei veiy simple example, analogous to an 
example pointed out by Schioder and by Tanneiy The expression 
(1 _- s R )/(l + s n ), where n is a positive integer which increases in- 
definitely, approaches the limit +1 if |*|< 1, and the limit 1 
if |*| > 1 If |*| = 1, this expiession has no limit except foi z = l 
Now the sum of the first n terms of the series 



is equal to the piecedmg expression This series is therefore conver- 
gent if || is different from unity Hence it repiesents + 1 in the 
interior of the ciicle C with the radius unity about the origin as 
centei, and 1 at all points outside of this circle. Now let/(s), 
$(z) be any two analytic functions whatever, for example, two 
integial functions Then the expiession 

1 

'2 J 

is equal to f(z) in the interior of C, and to < (z) in the region ex- 
tenor to C. The circumference itself is a cut for that expression, but 
of a quite different natuie from the natural boundaries which we 
have just mentioned. The function which is equal to $(&) in the 
interior of C can be extended analytically beyond C , and, similarly, 
the function which is equal to \l/(&) outside of C can be extended 
analytically into the interior. 

Analogous singularities present themselves in the case of functions 
represented by definite integrals The simplest example is furnished 
by Cauchy's integral, if/() is a function analytic within a closed 
curve F and also on that curve itself, the integral 

/_ 



v -} C*- 
V2Wj r *-x 

represents f(x) if the point x is in the interior of T The same inte- 
gral is zero if the point x is outside of the curve T, for the function 
/()/( x) is then analytic inside of the curve Here again the 
curve T is not a natural boundary for the definite integral. Similarly, 
the definite integral ^ 2ff ctn [(* se)/2]<fc has the real axis as a cut ; 
it is equal to + 2 iri or 2 TTI, according as x is above or below that 
cut ( 45). 



IV, 90] NATURAL BOUNDARIES CUTS 215 

90 Semite's formula An interesting result due to Hermite can be brought 
into i elation with the preceding discussion * Let F(t, z), G (t, z) be two analytic 
functions of each of the variables t and z , for example, two polynomials or two 
power &enes convergent foi all the values of these two vanables Then the 
definite integral 

"" 



taken over the segment of a straight line which joins the two points a and j3, 
represents, as we shall see later ( 95), an analytic function of z except for the 
values of z which are roots of the equation G- (t, z) = 0, wheie t is the complex 
quantity corresponding to a point on the segment a/3 This equation theiefoie 
determines a finite or an infinite number of cuives foi which the mtegial $(z) 
ceases to have a meaning Let AB be one of these curves not having any double 
points In older to consider a veiy precise case, we shall suppose that when t 
de&ciibes the segment #, one of the roots of the equation G(t, 2) = describes 
the aic AB, and that all the other roots of the same equation, if there are any, 
remain outside of a suitably chosen closed curve sui rounding the arc AB, so 
that the segment <ar/3 and the aic AB coirespond to each other point to point 
The integral (16) has no meaning when z falls upon the aic AB , we wish to 
calculate the difference between the values of the function $ (z) at two points 
JT, N'i lying on opposite sides of the arc AB, whose distances from a fixed point 
M of the arc AB are infinitesimal Let & + <?, + <' be the thiee values of z 
corresponding to the thiee points Jf, 
N, N' respectively To these thiee 
points coirespond in the plane of the 
vanable t, by means of the equation 
G (, z) = 0, the point m on <arj3, and 
the two points n, n" on opposite sides 
of a$ at infinitesimal distances fiom 
m Let 0, 6 + v, Q + if be the coi- 
le&pondmg values of t In the neighborhood of the segment aft let us take 
a point 7 so near aft that the equation G (, + c) = has no other root 
than t = Q H- 1? in the interior of the triangle afiy (Fig 35) The function 
Ffa f + )/<? (^ ^ 4. e ) of the variable t has but a single pole 6 + ijm the inteuor 
of the triangle or/37, and, according to the hypotheses made above, this pole 
is a simple pole Applying Cauchy's theorem, we have, then, the relation 



(17) 




T 



The two integrals f, f y " are of the same form as $(2) , they represent re- 
spectively two functions, ^(z), < 2 (2), which are analytic so long as the variable 
is not situated upon certain curves Let AC and BC be the curves which cor- 
respond to the two segments #7 and fty of the t plane, and which are at 
infinitesimal distances from the cut AB associated with < (z) Let us now give 

* HERMITE, Sur quelqites points de la theorie des fonetions (Crelle's Journal, 
Vol XCI) 



216 ANALYTIC EXTENSION [IV, 90 

the value f -f e' to z , the coriespondmg value of t is 6 + 17', represented by the 
point TI', and the function F(t, f + J)/G(t, + O of * 1S analytic m the interior 
of the triangle apy We have, then, the relation 



G+) * ffftr+O y <*,*+ 

g the two formulse (17) and (18) term by term, we can 
as follows 



<*,*+ 

subtracting the two formulse (17) and (18) term by term, we can wnte the result 



But since neithei of the functions ^(z), $ 2 (z) has the line AB as a cut, they 
are analytic m the neighboihoocl of the point z = f, and by making e and e' ap- 
proach zeio we obtain at the limit the difference of the values of $(z) in two 
points infinitely near each other on opposite sides ol AB We shall wnte the 
result in the abridged form 

(19) *w-*w = toSt&, 

30 

this is Hermite's formula It is seen that it is very simply related to Cauchy's 
theorem * The demonstration indicates clearly how we must take the points N 
and N' , the point N(f + e) must be such that an observer descubmg the segment 
a$ has the corresponding point 9 + 17 on his left 

It is to be noticed that the arc AB is not a natural boundary for the 
function $(z). In the neighborhood of the point JV' we can replace *(z) by 
[$!() + $ 2 ( z )] according to the relation (18) Now the sum ^(z) H- $ 2 ( z ) 1S 
an analytic function in the curvilinear triangle A CB and on the arc A B itself, 
as well as in the neighborhood of N'. Theiefore we can make the variable z 
cross the arc AB at any one of its points except the extremities A and B 
without meeting any obstacle to the analytic extension. The same thing would 
be true if we were to make the variable z cross the arc AB in the opposite sense 

Example. Let us consider the integial 



where the Integral is to be taken over a segment AB of the real axis, and where 
f(t) denotes an analytic function along that segment AB Let us represent z 
on the same plane as t . The function $ (z) is an analytic function of z in the 
neighborhood of every point not located on the segment AB itself, which is a 
cut for the integral The difference * (N) * (N*) is here equal to 2 m/(fl, 
where f is a point of the segment AB When the variable z crosses the line A B, 
the analytic extension of $ (z) is represented by * (z) 2 mf(z) 

This example gives rise to an important observation The function $ (z) is 
still an analytic function of z, even when/(4) is not an analytic function of , 
provided that f(t) is continuous between a and p ( 33) But m this case the 
preceding reasoning no longer applies, and the segment AB is in general a 
natural boundary for the function 



* GOURSA.T, Sur un tMortme de M Herrmte (Acta mathematica, Vol I) 



IV, Exs ] EXERCISES 217 

EXERCISES 

1. Pmd the lmes.of discontinuity for the definite integrals 



taken along the stiaight line which joins the points (0, 1) and (ot, b) respec- 
tively , determine the value of these integrals f 01 a point z not located on these 
boundaries 

2 Consider f om circles with radii 1/V2, having for centers the points + 1, 
-f i, 1, i The region exterior to these foui circles is composed of a finite 
region A^ containing the origin, and of an infinite region A 2 Construct, by the 
method of 38, a series of rational functions which converge in these regions, 
and whose value in A l is equal to 1 and in A 2 to Verify the result by finding 
the sum of the series obtained 

3 Treat the same questions, considenng the two regions interior to the circle 
of radius 2 with the center for origin, and exterior to the two circles of radius 1 
with centers at the points + 1 and 1 respectively 

[APPELL, Acta mathematica, Vol. I.] 

4 The definite integral 

fame 



taken along the real axis, has for cuts the straight lines x = (2 k -f 1) ?r, where k 
is an integer Let = (2 k + 1) TT + i be a point on one of these cuts The dif- 
feience in the values of the integial in two points infinitely close to that point 
on each side of the cut is equal to ie (erf + e-). 

[HERMITE, Crelle's Journal, Vol XCI ] 
5. The two definite integrals 

j C Ld, J a = C *^-i dt. 

J-o, Z J_ * 

taken along the leal axis, have the axis of reals for a cut in the plane of the 
variable z Above the axis we have J 2 TTI, J 9 = 0, and below we have J = 0, 
J Q s= 2m From these lesults deduce the values of the definite integrals 

+ a* 

+ 00 Q It 



'***** 

L, ^j=-. 



*- - - 

[HERMITE, CreZZe's Journal, Vol XCI ] 
6 Establish by means of cuts the formula (Chap. II, Ex 15) 
-.+ git ir 



'Consider the integral 

"*" - " fl+-a ri 



-oo 1+e* 

[EERMITE, Cretins Journal, Vol XCI,] 



218 ANALYTIC EXTENSION [IV, Exs 

which has all the stiaight lines y = (2 k -f 1) TT f or cuts, and which remains con- 
stant in the strip included between two consecutive cuts Then establish the 
relations 



where 2; and z 4- 2 TTI are two points separated by the cut y TT ) 

7*. Let/(2) be an analytic function in the neighborhood of the origin, so that 
f(z) =Sa n z n Denote by F(z) =SOn2 n / n ' the associated integral function It is 
easily proved that we have 

(1) 

^ ' 

where the integral is taken along a closed curve C, including the origin within 
it, inside of which f(z) is analytic From this it follows that 



r l _, , 1 rf(u) 
(2) / e-*F(az)da = - \ J -^- 

x ' Jo 2m Jc u 



where I denotes a real and positive number 

If the leal part ot z/u remains less than 1 e (where e> 0) when u describes 
the curve (7, the integral 



C l 

I 

Jo 



e*V Vda 
o 

approaches u/(u s) uniformly as Z becomes infinite, and the formula (2) be- 
comes at the limit 



rv^(a*x*a=-i- c m^ = 

Jo v ' Zm J?) u z 



This result is applicable to all the points within the negative pedal curve of C 

[BOREL, Lemons sur les series divergences ] 

8*. Let/(2) = SOn2 n , ^> (2;) = S&nZ" be two power series whose radii of conver- 
gence are r and p respectively The series 



has a radius of convergence at least equal to rp, and the function ^ (z) has no 
other singular points than those which are obtained by multiplying the quanti- 
ties coriesponding to the different singular points off(z) by those corresponding 
to the singular points of $ (z) 

[HADAMABD, Ada mathematica, Vol. XXIII, p 55 ] 



CHAPTER V 
ANALYTIC FUNCTIONS OF SEVERAL VARIABLES 

I GENERAL PROPERTIES 

In this chapter we shall discuss analytic functions of several 
independent complex vanables For simplicity, we shall suppose 
that there aie two variables only, but it is easy to extend the results 
to functions of any number of variables whatever 

91 Definitions Let & = u -f- vi, l = w -f ti be two independent 
complex variables , every other complex quantity Z whose value 
depends upon the values of & and & 1 can be said to be a function of 
the two variables & and #' Let us represent the values of these two 
variables & and 2' by the two points with the coordinates (u, v) and 
(w, t) in two systems of rectangular axes situated in two planes P, P', 
and let A, A 1 be any two portions of these two planes We shall say 
that a function Z =/(#, #') is analytw in the two regions A, A' if 
to every system of two points , ', taken respectively in the regions 
A t A', corresponds a definite value of f(z, #'), varying continuously 
with & and 2', and if each of the quotients 

/(* + *,)-/(*,') /(*,*' + &)~/0,* f ) 

h ' k 

approaches a definite limit when, and #' remaining fixed, the 
absolute values of h and k approach zero These limits are the 
partial derivatives of the function /(#, '), and they are represented 
by the same notation as in the case of real variables 

Let us separate in /(#, #') the real part and the coefficient of i } 
f(z y #') = X -f Yi ; X and Y are real functions of the four independ- 
ent real variables u, v, w, t, satisfying the four relations 

&*T__aF 2____F ? = ?Z = _Z 

0tt ~~ dv ' 8v ~ du,' dw ~~ dt ' fa ~~ dw' 

the significance of which is evident.* We can eliminate T in six 

* If z and tf are analytic functions of another variable a:, these relations enable us 
to demonstrate easily that the derivative of /(z, "wrtk respect to as is obtained by the 
usual rule which gives the derivative of a function of other functions The formulae 
of the differential calculus, in particular those for the change of variables, apply, 
therefore, to analytic functions of complex variables 

219 



220 SEVERAL VARIABLES [V,91 

different ways by passing to derivatives of the second order, but 
the six relations thus obtained i educe to only four 

(J^^JjL^O d * X &X = Q 

dudt dvdw ' dudw dvdt ' 

d*X PX _ VX d^X 

Jtf + ~W " ' dw* + 9f ~~ 

Up to the present time little use has been made of these relations 
for the study of analytic functions of two vanables One reason for 
this is that they are too numeious to be convenient 

92. Associated circles of convergence. The properties of power series 
in two real variables (I, 190-192, 2d ed. ; 185-186, 1st ed ) are 
easily extended to the case where the coefficients and the variables 
have complex values Let 



(2) F(z,z') 

be a double series with coefficients of any kind, and let 



We have seen (I, 190, 2d ed) that theie exist, in general, an 
infinite number of systems of two positive numbers R 9 R ! such that 
the series of absolute values 

(3) 2A mn ZZ' n 

is convergent if we have at the same time Z<R and Z'<R', and 
divergent if we have Z>R and Z'>R' Let C be the cucle de- 
scribed in the plane of the variable # about the origin as center with 
the radius R ; similarly, let C 1 be the circle described in the plane of 
the variable #' about the point #' = as center with the radius R 1 
(Fig. 36) The double series (2) is absolutely con vei gent when the 
variables s and #' are respectively in the interior of the two cucles C 
and C', and divergent when these variables are respectively extenor to 
these two circles (I, 191, 2d ed ; 185, 1st ed ) The circles C, C" 
are said to form a system of associated circles of convergence This 
set of two circles plays the same part as the circle of convergence 
for a power series in one variable, but in place of a single circle 
there is an infinite number of systems of associated circles for a 
power series in two variables. Tor example, the series 



ml n\ 



v, 



GENERAL PROPERTIES 



221 



is absolutely convergent if |g|-f |2 r |<l, and in that case only 
Every pair of circles C, C" whose radii R, R' satisfy the relation 
R -f R* = 1 is a system of associated circles. It may happen that we 
can limit ourselves to the consideration of a single system of asso- 
ciated circles , thus, the series S2 m 2'* is convergent only if we have 
at the same time \z\ < 1 and |#'| < 1 

Let C l be a circle of ladius R % <R concentric with C; similarly, 
let Ci be a circle of radius Ri<R' concentric with C", when the 
variables 2 and 2' remain within the circles C^ and C[ respectively, 




the series (2) is uniformly convergent (see I, 191, 2d ed , 185, 
1st ed ) and the sum of the series is therefore a continuous function 
F(z, 2') of the two variables 2, 2' in the interioi of the two circles 
C and C". 

Differentiating the series (2) term by term with respect to the 
variable 2, for example, the new series obtained, ^ma mn z m " 1 z' n } is again 
absolutely convergent when 2 and 2' lemain in the two circles C and 
C" respectively, and its sum is the denvative dF/dz of F(z, 2') with 
respect to 2 The proof is similar in all respects to the one which has 
been given for real variables (I, 191, 2d ed , 185, 1st ed.) Simi- 
larly, jF(2, 2') has a partial derivative dF/dz' with respect to 2', which 
is represented by the double series obtained by differentiating the 
series (2) term by term with respect to 2 r The function F(z, 2 r ) is 
therefore an analytic function of the two variables 2, # f in the pre- 
ceding region The same thing is evidently true of the two deriva- 
tives 8F/dz, dF/8z', and therefore F(z, 2^ can be differentiated term 



222 



SEVERAL VARIABLES 



[V, 92 



by term any number of times ; all its partial derivatives are also 
analytic functions 

Let us take any point of absolute value r in the interior of C, and 
from this point as center let us describe a circle c with ladius R r 
tangent internally to the circle C In the same way let z' be any point 
of absolute value r 1 < It', and c' the circle with the point z 1 as center 
and R 1 r 1 as ladius Finally, let z + h and z' 4- k be any two points 
taken in the cncles c and c' respectively, so that we have 

|*| + |A|<jR, \z'\ + \k\<R' 

If we replace z and z 1 in the series (2) by z + h and z 1 + k, we can 
develop each term in a senes proceeding accoiding to poweis of h 
and k, and the multiple senes thus obtained is absolutely con vei gent 
Arranging the senes according to powers of h and k, we obtain the 
Taylor expansion 



(*) 



93 Double integrals. When we undertake to extend to functions 
of several complex variables the general theorems which Cauchy 
deduced from the consideration of definite integrals taken between 
imaginary limits, we encounter difficulties which have been com- 
pletely elucidated by Poincare * We shall study here only a very 






w 



FIG 37 

simple particular case, which will, however, suffice for our subse- 
quent developments. Let /(, *) be an analytic function when the 
variables #, z 1 remain within the two regions A } A' respectively 
Let us consider a curve ah lying in A (Fig 37) and a curve a!V 
in A\ and let us divide each of these curves into smaller arcs by 
any number of points of division. Let , #1, 2 , , ^-i? ***> > z 



* PoiffCAB6, Sur ks res^dus ties integrates doubles (Acta mattemahca, Vol IX), 



V,93] GENERAL PROPERTIES 223 

be the points of division of ab, where Z Q and Z coincide with a and b, 
and let z' Q , &{, z' 2 , , ' h _ l9 z' h , , *m-u z * ^ e tne points of division 
of a'b', where J and Z 1 coincide with a' and V The sum 

(5) s=j? V /( St _ u 4.0 (* - % _ x ) - ,,'.0, 

11 A=l 

taken with respect to the two indices, approaches a limit, when the 
two numbers m and n become infinite, 111 such a way that the abso- 
lute values \z k ^_ T | and |i 4-i| appioach zero. Let f(z, ') 
= X+Yi, wheie X and Y aie real functions of the four vanables 
u, v, w, t, and let us put k = % -j- # A fc, *j[ = u\ + t h i The general 
term of the sum S can be wntten in the form 



Xi( k _ ly v k _^ w h _ l} t h + 

X [% %-i 4- t(^ t *JL- 

and if we cany out the indicated multiplication, we have eight 
partial pioducts Let us show, for example, that the sum of the 
partial pioducts, 



approaches a limit We shall suppose, as is the case in the figure, 
that the curve ab is met in only one point by a parallel to the axis Ov, 
and, siniilaily, that a parallel to the axis Ot meets the curve a'b* in at 
most one point Let v = <(w), t = ty(w) be the equations of these 
two curves, U Q and U the limits between which u vanes, and W Q and 
W the limits between which w varies. If we replace the variables v 
and t in X by <(w) and $(w) respectively, it becomes a continuous 
function P(u, w) of the vanables u and w, and the sum (6) can again 
be wntten in the fonn 

n m 

(6 f ) 

As m and TI become infinite, this sum has for its limit the double 
integral ffPfyt, w)dudw extended over the rectangle bounded by the 
straight lines u = , u = C7", w = w , w = W. 

This double integral can also be expressed in the form 



/> u /*w 

I du I P(u y 

Ju n Jw n 



224 SEVERAL VARIABLES [V,93 

or again, by introducing line integrals, in the form 
(7) C du I X(u, v, w, t)dw. 

J(ab) J(a'b') 

In this last expression we suppose that u and v are the coordinates 
of any point of the arc ab, and w y t the coordinates of any point of 
the arc a'b'. The point (u, v) being supposed fixed, the point (w, t) 
is made to descube the aic a'b', and the line integral fXdw is taken 
along a'b'. The result is a function of w, v, say R (u, v) , we then 
calculate the line integral/is (u, v) du along the arc ab 

The last expiession (7) obtained for the limit of the sum (6) is 
applicable whatever may be the paths ab and a'b'. It suffices to break 
up the arcs ab and a'b' (as we have done repeatedly before) into 
arcs small enough to satisfy the previous requirements, to associate 
in all possible ways a portion of ab with a portion of a'b', and then 
to add the results Proceeding in this way with all the sums of par- 
tial products similar to the sum (6), we see that S has for its limit 
the sum of eight double integrals analogous to the integral (7) 
Representing that limit byJJjF(#, z'^dzdz', we have the equality 

')dzdz'= i du C Xdw \ dv \ Xdt 

J(ab) J(a'b') /(&) /(a'&') 

- C du C Ydt - i dv C Ydw 

J(ab) */('&') c/(a&) t/(a'&') 

+ i C du C Ydw ~i I dv C Ydt 

J(a&) J(a'b') /(a&) */(a'&') 

+ i I du I Xdt +i I dv I Xdw, 

/(<*&) t/(a'&') */(&) J(a'b') 

which can be written in an abridged form, 

ffp(z, z 1 ) dzdz' = r (du + idv) C (X + <F) (dw + idf), 

JJ */(&) c/O'&') 



(8) 



/(&) 

or, again, 



(9) ffffa z^dzdz' = C ds C F(z> s') dd 

JJ t/(o&) J(a'b') 

The formula (9) is precisely similar to the formula for calculating 
an ordinary double integral taken over the area of a rectangle by 
means of two successive quadratures (I, 120, 2d ed. ; 123, 1st ed ). 
We calculate first the integral fF(z, z'} dz' along the arc a'b', supposing 



V,94] GENERAL PROPERTIES 225 

2 constant, the result is a function $(2) of 2, which, we integrate 
next along the arc db As the two paths db and a)V enter in 
exactly the same way, it is clear that we can interchange the order 
of integrations 

Let M be a positive number greater than the absolute value of 
F(z, z 1 ) when & and z 1 descube the arcs db and a)V If L and L' 
denote the lengths of the respective arcs, the absolute value of the 
double integral is less than MIL' ( 25) When one of the paths, a'b' 
for example, forms a closed curve, the integral f^ a > bf) F(z y z')dz' will 
be zero if the function F(z, z'*) is analytic for all the values of z' in 
the interior of that cuive and for the values of z on db. The same 
thing will then be true of the double integral 

94. Extension of Cauchy's theorems. Let C, C 1 be two closed curves 
without double points, lying respectively in the planes of the variables 
z and z 1 , and let F(z, z') be a function that is analytic when z and z* 
remain in the regions limited by these two curves or on the curves 
themselves Let us consider the double integral 



I=| dz I ^ 9 *'\ r-j 

J(O J(co v 5 ~~ * A* ~~ ^ / 

where a; is a point inside of the boundary (7 and where x 1 is a point 
inside of the boundary C ! , and let us suppose that these two bound- 
aries are described in the positive sense The integral 

JP(g, z')dz' 



where & denotes a fixed point of the boundary (7, is equal to 
2 iri F(z, d)/( x) We have, then, 



or, applying Cauchy's theorem once more, 
/=: 4-77^(0;, a? f ) 
This leads us to the formula 



(10) 

^ ' 



is completely analogous to Caueliy's fundamental formula, and 
from which we can derive similar conclusions Erom it -we deduce 



226 SEVERAL VARIABLES [V, 94 

the existence of the partial derivatives of all orders of the function 
F(z, s') m the regions considered, the derivative d m+n F/dx m dx fn hav- 
ing a value given by the expiession 

S M+n F m* 



In order to obtain Taylor's formula, let us suppose that the 
boundanes C and C' are the circumferences of cncles Let a be the 
center of C, and R its radius, b the center of C", and R' its radius 
The points x and x 1 being taken respectively in the interior of these 
circles, we have \x a\ = r < R and \x' b \ = r 1 < R' Hence the 
rational fraction 



(* - oj)(*' -a; 1 ) [*--( a)][*' - 
can be developed in powers of x a and x 1 b, 



where the series on the right is uniformly conveigent when 2 and z f 
describe the circles C and C' respectively, since the absolute value of 
the general term is (r/R) m (r'/R') n /RR f We can theiefore replace 
!/( )(' x') by the preceding series in the relation (10) and 
integrate term by term, which gives 



Making use of the results obtained by replacing x and x' by a and b 
in the relations (10) and (11), we obtain Taylor's expansion in the 
form 



where the combination m = 71 = is excluded from the summation 
JVbte. The coefficient a^ of (as a) m (x' 5) n in the preceding 
series is equal to the double integral 



'_ 7>Y + i 



V,95] GENERAL PROPERTIES 227 

If M is an upper bound foi | F(z, ')| along the cncles C and C.', we 
have, by a previous general remark, 



The function 

M 



is therefore a dominant function for F(x } #') (I, 192, 2d ed ? 
186, 1st ed ) 

95. Functions represented by definite integrals. In order to study 
certain functions, we often seek to express them as definite integrals 
in which the independent variable appears as a parameter under the 
integral sign We have already given sufficient conditions under 
which the usual rules of differentiation may be applied when the 
variables are real (I, 98, 100, 2d ed , 97, 1st ed.). We shall 
now reconsider the question for complex variables 

Let F(z, #') be an analytic function of the two variables & and & f 
when these variables remain within the two regions A and A 1 respec- 
tively. Let us take a definite path L of finite length in the region A, 
and let us consider the definite integral 



(13) *(*)= f F(z,x)dz, 

J(Ly 



where x is any point of the region A 1 . To prove that this function 
< (x) is an analytic function of x 9 let us describe about the point x as 
center a circle C with radius R, lying entirely in the region A 1 Since 
the function F(z, '} is analytic, Cauchy's fundamental formula gives 



whence the integral (13) can be written in the form 

1 



' x 

Let x + Ax be a point near x in the circle C ; we have, similarly,, 

F(z,z')dz' 



228 SEVERAL VARIABLES [V, 95 

and consequently, by repeating the calculation already made ( 33), 

F (*, 



A* - 2** JnJto (*'-*? 

Aar F(, 



Let M be a positive number greater than the absolute value of 
F(, &') when the variables and s* describe the curves L and C 
respectively , let S be the length of the cui ve L , and let p denote the 
absolute value of Ax. The absolute value of the second integral is 
less than 

M 



hence it approaches zero when the point a + Ax approaches x in- 
definitely It follows that the function < (a?) has a unique derivative 
which is given by the expiession 

_. ,, , i r 7 r F (*, 

*'(7!)=- - I dz \ -j-f 
V ' 2 ^J(L) J(P) (*' 

But we have also ( 33) 



and the preceding relation can be again wiitten 
*'()= I ^-dz. 



Thus we obtaui again the usual formula for differentiation under the 
integral sign 

The reasoning is no longer valid if the path of integration L 
extends to infinity. Let us suppose, for definiteness, that i is a 
ray proceeding from a point a Q and making an angle & with the 
real axis. We shall say that the integral 



$(x)= I F(z, x)dz 

is uniformly convergent if to every positive number e there can 
be made to correspond a positive number N such that we have 

JF(,aj)<b 



V,96] GENERAL PROPERTIES 229 

provided that p is greater than N 9 wherever x may be in A 1 By 
dividing the path of integration into an infinite number of recti- 
lineal segments we piove that every uniformly convergent integial 
is equal to the value of a uniformly convergent series whose teims 
are the integrals along certain segments of the infinite lay L. All 
these integrals are analytic functions of x, therefore the same is 
tiue of the integral f~F(z, x)d* ( 39) 

It is seen, in the same way, that the ordinary formula for differen- 
tiation can be applied, provided the integral obtained, ^(dF/dx)dz, 
is itself uniformly convergent. 

If the function F(z, 2') becomes infinite for a limit # of the path 
of integration, we shall also say that the integral is uniformly con- 
vergent in a certain region if to every positive number e a point 
a Q + TI on the line L can be made to correspond in such a way that 



I/' *,.> 

I */a + Tj 



where 5 is any point of the path L lying between a Q and a Q -f 77, the 
inequality holding for all values of x in the region considered. 
The conclusions aie the same as in the case where one of the limits 
of the integral is moved off to infinity, and they aie established in 
the same way 

96 Application to the T function The definite integral taken along the real axis 
(15) T(2) 



which we have studied only for real and positive values of z (I, 94, 2d ed ; 
92, 1st ed ), has a finite value, provided the real part of z, which we will denote 
by *R(), is positive. In fact, let z = x + yi, this gives \t 9t - 1 
Since the integral +CD 

i t 
JQ 

has a finite value if x is positive, it is clear that the same is true of the integral 
(15) (I, 91, 92, 2d ed , 90, 91, 1st ed.) This integral is uniformly con- 
vergent m the whole region defined by the conditions N> < R(z)>i], where N 
and 7j are two arbitrary positive numbers In fact, we can write 



r(z)= f V- 

/0 



and it suffices to prove that each of these integrals on the right is uniformly 
convergent Let us prove this for the second integral, for example. Let I be a 
positive number greater than one If *% (z) < N, we have 



ir*- 



230 SEVERAL VARIABLES [V, 96 

and a positive number A can be found laige enough to make the last integral 
less than any positive number e whenevei I ^ A The function r (z), defined by 
the integral (15), is theiefoie an analytic function m the whole legion of the 
plane lying to the light of the ^-axis This function r (z) satisfies again the 
relation 



(16) 

obtained by mtegiation by parts, and consequently the more general relation 
(17) r(z + n) = z(z + l) (z + n-l)r(z), 

which is an immediate consequence of the other 

This piopeity enables us to extend the definition of the T function to values 
of z whose real part is negative For consider the function 



where n is a positive integer The numeiator r (z + n) is an analytic function 
of z defined for values of z for which ft (z) > n , hence the function ^ (z) is a 
function analytic except for poles, defined for all the values of the variable 
whose real part is greater than n Now this function ^ (z) coincides with the 
analytic function r (z) to the light of the y-axis, by the relation (17), hence it 
is identical with the analytic extension of the analytic function F (z) in the 
strip included between the two stiaight lines ^(z) = 0, ft(z) = n Since the 
number n is arbitraiy, we may conclude that there exists a function which is 
analytic except for the poles of the first order at the points z = 0, z = 1, 
z = 2, -, z = n, - , and which is equal to the integral (15) at all points to 
the right of the y-axis This function, which is analytic except for poles in the 
finite plane, is again represented by r (2) , but the formula (15) enables us to 
compute its numerical value only if we have ^ (z) > If ^ (z) < 0, we must also 
make use of the i elation (17) in order to obtain the numerical value of that 
function 

We shall now give an expression for the r function which is valid for all 
values of z Let S(z) be the integral function 

S(z) = zl 

n- 

which has the poles of r (z) for zeros The product 8 (z) T (z) must then be 
an integral function. It can be shown that this integral function is equal to 
6-0% where <7 is Euler's constant* (I, 18, Ex., 2ded , 49, Note, 1st ed ), 
and we denve from it the result 



(19) 



which shows that 1/T (z + 1) is a transcendental integral function. 



HBEMIMI, Cours ^Analyse, 4th ed , p 142 



V,97] GENERAL PROPERTIES 231 

97, Analytic extension of a function of two vanables. Let u = F(z, tf) be an 
analytic function of the two variables z and z f when these two variables remain 
respectively in two connected regions A and A! of the two planes in which we 
lepresent them It is shown, as m the case of a single variable ( 83), that the 
value of this function foi any pair of points 2, z' taken in the regions A, A' is 
detei mined if we know the values of F and of all its partial derivatives for a 
pair of points z = a, z f = o taken in the same regions It now appears easy to 
extend the notion of analytic extension to functions of two complex variables 
Let us consider a double series Sa mn such that there exist two positive numbers 
7, having the following property the series 

(20) F(z,z^) = ^a mn z^^ 

is convergent if we have at the same time |s| < r, |z' | < r', and divergent if we 
have at the same time \z\ > r, \i\ > r' The preceding series defines, then, a 
function F(z, z') which is analytic when the vanables z, zf remain lespectively 
in the circles 0, C' of radii r and K , but it does not tell us anything about the 
natuie of this function when we have \z\>r or \z f \>r" Let us suppose for 
defimteness that we cause the variable z to move over a path L from the origin 
to a point Z exterior to the circle C, and the variable f to travel over another 
path L' from the point z' = to a point Z' exterior to the circle C" Let a and 
ft be two points taken respectively on the two paths and I/, a being in the 
intenor of C and ft m the interior of C' The series (20) and those which are 
obtained from it by successive differentiations enable us to form a new power 
senes, 

(21) 



which is absolutely convergent if we have \z a \ < r t and [zf ft \ < r^ where 
r x and r{ are two suitably chosen positive numbeis Let us call C l the circle of 
radius r t described about the point a as center in the plane of z, and C{ the 
circle of radius r{ described in the plane of z f about the point ft as center If z 
is in the part common to the two circles and C 1? and the point z* m the part 
common to the two circles C' and C^ the value of the series (21) is the same as 
the value of the series (20) If it is possible to choose the two numbers r t and r{ 
in such a way that the circle (7, will be partly exterior to the circle (7, or the 
circle G{ partly exterior to the circle C', we shall have extended the definition 
of the function F(z, z') to a region extending beyond the first Continuing in 
this manner, it is easy to see how the function F(z, zf) may be extended step by 
step. But there appears here an important new consideration : It is necessary 
to take into account the way in which the variables move unth respect to each other 
on their respective paths The following is a very simple example of this, due to 
Sauvage * Let u = "Vz zf 4- 1 , for the initial values let us take z=:z' = Q,u=:l, 
and let the paths described by the variables 2, z* be defined as follows . 1) The 
path described by the variable z f is composed of the rectilinear segment from 
the origin to the point z* = 1 2) The path described by z is composed of three 
semi circumferences the first, OM A (Fig. 38), has its center on the real a:ns to 

* Premiers pnneipes de la the'one generate tie* fonctwns de plusieurs variables 
(Annales de la Faculte" ties Sciences de Marseille, Vol. XIV) This memoir is an 
excellent introduction to the study of analytic functions of seveial variables 



232 



SEVERAL VARIABLES 



[V, 97 



the left of the origin and a ladius less than 1/2 , the second, ANB, also has its 
center on the real axis and is so placed that the point 1 is on its diameter AB , 
finally, the third, BPC, has for its centei the middle point of the segment joining 
the point B to the point C(z = 1) The fiist and the thud of these seinicircum- 
ferences are above the leal axis, and the second is below, bo that the bound- 
ary OMANBPCO incloses the point z = 1 Let us now select the following 
movements 

1) tf remains zero, and z descubes the entiie path OABC , 

2) z lemams equal to 1, and z f descubes its whole path 

If we consider the auxihaiy vanable t = z z', it is easily seen that the path 
described "by the variable , when that variable is represented by a point on the 




. 38 



2 plane, is precisely the closed boundary OABCO which surrounds the critical 
point t = 1 of the radical V$ + 1. The final value of u is therefore u = 1 
On the other hand, let us select the following procedure 

1) z remains zero and z' varies from to 1 e (e being a very small positive 
number) ; 

2) z' remains equal to 1 e, and z describes the path OABC , 

3) z remains equal to 1, and z* varies from 1 e to 1 

When zf varies from to 1 e, the auxiliary variable t descubes a path 00' 
ending in a point 0' very near the point 1 on the real axis When z describes 
next the path OABC, t moves over a path (/A'tfO' congruent to the preceding 
and ending in the point C' '(OC' = e) on the real axis Finally, when sf varies 
from 1 e to 1, t passes from C" to the origin. Thus the auxiliary variable t 
describes the closed boundary OO'A'B'C'O which leaves the point 1 on its 
exterior, provided e is taken small enough. The final value of u will therefore 
be equal to + 1. 

Very much less is known about the nature of the singularities of analytic 
functions of several variables than about those of functions of a single variable 
One of the greatest difficulties of the problem lies m the fact that the pairs of 
singular values are not isolated * 



*For everything regarding this matter see a memoir by Pomcarl in the Acta 
mathematica (Vol X2CVT), and P Cousin's thesis (Ibid Vol XIX) 



V,98] IMPLICIT FUNCTIONS 233 

n IMPLICIT FUNCTIONS ALGEBRAIC FUNCTIONS 

98. Weierstrass's theorem. We have already established (I, 193, 
2d ed , 187, 1st ed ) the existence of implicit functions defined by 
equations in which the left-hand side can be developed m a power 
series proceeding in positive and increasing powers of the two 
variables The arguments which were made supposing the variables 
and coefficients real apply without modification when the variables 
and the coefficients have any values, real or imaginary, provided we 
retain the other hypotheses We shall establish now a more general 
theorem, and we shall preserve the notations previously used in that 
study. The complex variables will be denoted by x and y. 

Let F(x, y) be an analytic function in the neighborhood of a 
pair of values x = a, y = /3, and such that we have F(a, /?) = 0. 
We shall suppose that a = ft 0, which is always permissible The 
equation F(Q, ?/) = has the root y = to a certain degree of mul- 
tiplicity. The case which we have studied is that in which y = is 
a simple root , we shall now study the geneial case where y = is a 
multiple root of order n of the equation .F(0, y) = 0. If we ai range 
the development of F(x, y) in the neighborhood of the point x = y = 
accoidmg to poweis of y, that development will be 

(22) F(x, y^^Aq+Aj/ + -\~A n y n i-A n + l y n+1 -f * > 

where the coefficients A % are power series in x, of which the first n 
are zero for x = 0, while A n does not vanish for x = Let C and C' 
be two circles of radii R and R' described in the planes of x and y 
respectively about the origin as center. We shall suppose that the 
function F(x, y) is analytic in the region defined by these two circles 
and also on the circles themselves , since A n is not zero for #=0, we 
may suppose that the radius R of the circle C is sufficiently small 
so that A n does not vanish in the interior of the circle C nor on the 
circle Let M be an upper bound for | F(x, y) \ in the preceding region 
and B a lower bound foi \A n \ By Cauchy's fundamental theorem 
we have 



where x and y are any two points taken in the circles C and C f ; 
from this we conclude that the absolute value of the coefficient A m 
of y m in the formula (22) is less than M/R' m } whatever may be the 
value of x in the circle C 



234 SEVERAL VARIABLES [V, 98 

We can now write 
(23) *(,?) 

where 



Let p be the absolute value of y , we have 



j 

BR"\S' R' 2 I BR* 

R' 

and this absolute value will be less than 1/2 if we have 



On the other hand, let /-c(r) be the maximum value of the absolute 
values of the functions A Q , A v , A n _ l for all the values of x for 
which the absolute value does not exceed a number r < R Since 
these n functions are zero for x = 0, p (r) approaches zero with r, 
and we can always take r so small that 



where p is a definite positive number The numbers r and p having 
been determined so as to satisfy the preceding conditions, let us re- 
place the circle C by the circle C r described in the as-plane with the 
radius r about the point x = as center, and similarly in the y-plane 
the circle C' by the concentric circle C' p with the radius p. If we give 
to x a value such that |#|=i r, and then cause the variable y to 
describe the circle C'p, along the entire circumference of this circle we 
have, from the manner in which the numbers r and p have been chosen, 
|P| < 1/2, | Q| < 1/2, and therefore |P + Q| < 1. If the variable y 
describes the circle Cp in the positive sense, the angle of 1 + P + Q 
returns to its initial value, whereas the angle of the factor A^f in- 
creases by 2 nir The equation F(x 9 y) = 0, in which | x | ^ r, therefore 
has n roots whose absolute values are less than p, and only n, 

All the other roots of the equation F(x, y) = 0, if there are any, 
have their absolute values greater than p Since we can replace the 
number p by a number as small as we wish, less than p, if we replace 



V, 98] IMPLICIT FUNCTIONS 235 

at the same time r by a smallei numbei satisfying always the con- 
dition (25), we see that the equation F(x, y) = has n roots and only 
n which appioach zero with x 

If the vaiiable x lemains in the interior of the cncle C, or on its 
circumference, the n roots y v y 2 , , y n , whose absolute values are less 
than p, remain within the circle C' p These loots are not in general 
analytic functions of x in the cncle C,, but every symmetric integral 
rational function of these n roots is an analytic function of x in this cn- 
cle It evidently suffices to prove this foi the sum y\ -f- y% 4- 4- ?/n> 
where k is a positive integer Let us consider for this purpose the 
double integial 



I 7,1 II 

*= I d y ] v' L 

*S(C) *J{CA 



where we suppose \x\ < r If y 1 = p, the function .F (#', ?/') cannot 
vanish for any value of the variable x 1 within or on C r , and the only 
pole of the function under the integral sign m the interior of the 
circle C r is the point x' = x We have, then, 



't/CCr} 



and consequently 



By a general theorem ( 48) this integral is equal to 




where ?/ 1? 2/ 2> , y n aie the n roots of the equation F(x, y) = with 
absolute values less than /> On the other hand, the integral / is an 
analytic function of x in the circle C r , for we can develop !/(#' or) 
in a uniformly convergent series of powers of x, and then calculate 
the integral term by term. The different sums 5?/? being analytic 
functions in the circle C r , the same thing must be true of the sum 
of the roots, of the sum of the products taking two at a time, and so 
on, and therefore the n roots y v y# , y n are also roots of an equar 
tion of the nth degree 

(26) 



286 SEVERAL VARIABLES [v, 98 

whose coefficients a v a 2 , , a n are analytic functions of x in the 
cncle C r vanishing for x = 0. 

The two functions F(x 9 y} and f(x 9 y) vanish f 01 the same pairs 
of values of the variables x, y in the interior of the circles C r and C' p 
We shall now show that the quotient F(x } y)/f(x, y) is an analytic 
function in this region Let us take definite values for these vari- 
ables such that |;c| O, \!/\<p, and let us consider the double 

integral 

r r . C F(x\ y 1 ) dot 

J = 



IFor a value of y 1 of absolute value p the function /(&', y') of the 
variable x' cannot vanish for any value of x 1 within or on the circle 
C r The function undei the integral sign has theiefore the single 
pole x' = x within C r} and the corresponding residue is 



Hence we have also 

J = 



but the two analytic functions F(x 3 ?/'), /(#, ?/) of the variable ?/ 
have the same zeros with the same degrees of multiplicity in the 
interior of C' p . Their quotient is therefore an analytic function of 
y ! in C' p , and the only pole of the function to be integrated in this 
circle is y' = y , hence we have 




On the other hand, we can replace !/(#' x) (y' y) in the inte- 
gral by a uniformly convergent series arranged in positive powers 
of x and y. Integrating term by term, we see that the integral is 
equal to the value of a power series pioceeding according to powers of 
x and y and convergent in the circles C t> C' p Hence we may write 



or 

(27) 

where the function H(x 9 y) is analytic in the circles C r , C' f . 

The coefficient A n of y in F(x, y) contains a constant term dif- 
ferent from zero , since a v a# - , a n are zero ^ or lffiy|^| e develop- 
ment of H(x y y) necessarily contains a constant tipf'i^prent from 
zero, and the decomposition given by the expre^(|^4l|^) throws 



v,99] IMPLICIT FUNCTIONS 237 

into relief the fact that the roots of F(x, y)= which approach zero 
with x aie obtained by putting the first factor equal to zero The 
preceding important theorem is due to Weierstrass.* It generalizes, 
at least as far as that is possible for a function of several variables, 
the decomposition into factors of functions of a single variable. 

99. Critical points. In order to study the n roots of the equation 
F(x, y) = which become infinitely small with x, we are thus led to 
study the roots of an equation of the form 



foi values of x near zero, where a^ a 2 , -, a n are analytic functions 
that vanish for x = When n is greater than unity (the only case 
which concerns us), the point x = is in geneial a critical point. Let 
us eliminate y between the two equations / = and df/dy = , the 
resultant A (x) is a polynomial in the coefficients a v a^ , a n , and 
therefore an analytic function in the neighborhood of the ongm. 
This resultant t is zeio for x = 0, and, since the zeros of an analytic 
function form a system of isolated points, we may suppose that we 
have taken the radius r of the circle C r so small that in the interior 
of C r the equation A (x) = has no other loot than x = 0. Tor every 
point X Q taken in that circle othei than the origin, the equation 
f( x o) 2/) = will have n distinct roots According to the case already 
studied (I, 194, 2d ed ; 188, 1st ed ), the n roots of the equation 
(28) will be analytic functions of x in the neighborhood of the point 
# Hence there cannot be any other critical point than the origin 
in the intenor of the encle C r . 

Let y t , y^ -, y n be the n roots of the equation /(cc , ?/) = 0. Let 
us cause the variable x to describe a loop around the point x = 0, 
starting from the point X Q , along the whole loop the n roots of the 
equation f(x, y) = are distinct and vary in a continuous mannei. 
If we start from the point X Q with the root y v for example, and fol- 
low the continuous variation of that root along the whole loop, we 
return to the point of departure with a final value equal to one of the 
roots of the equation /(cc , y) = If that final value is y v the root 

* Abhandlungen aus der Functionerilehre von K Wezerstrass (Berlin, 1860) The 
proposition can also be demonstrated by making use only of the properties of power 
series and the existence theorem for implicit functions (Bulletin de la Soat6 
mathematique, Vol XXXVI, 1908, pp 209-215) 

t We disregard the case where the resultant is identically zero In this case / (x, y) 
would be divisible by a factor [fi(x, y)]*, where k > 1, fi(z 9 y) being of the same 
form as /(a, y) 



238 SEVERAL VARIABLES [V,99 

consideied is single-valued in the neighborhood of the origin If 
that final value is different from y^ let us suppose that it is equal 
to y 2 . A new loop descubed in the same sense will lead from the 
root y 2 to one of the roots y v y^ - , y n The final value cannot be 
y^ since the reverse path must lead from y 2 to y^ That final value 
must, then, be one of the loots y^ y^- -, y n If it is y^ we see that 
the two roots y l and y 2 are permuted when the vanable describes 
a loop around the origin If that final value is not y^ it is one 
of the remaining (n 2) roots , let y s be that root A new loop 
descubed in the same sense will lead from the root y g to one of the 
loots y l9 y 2 , y 3 , y 4 , - , y n It cannot be y g , for the same reason as 
before , neither is it y 2 , since the reverse path leads from y 2 to y^ 
Hence that final value is either y l or one of the remaining (n 3) 
roots y^ y 5 , , y n If it is y 1? the three loots y v y 2 , y z permute 
themselves cyclically when the variable x describes a loop around 
the origin, If the final value is different from y^ we shall continue 
to cause the vanable to turn around the origin, and at the end of 
a finite number of operations we shall necessarily come back to a 
root already obtained, which will be the root y l Suppose, for exam- 
ple, that this happens after p operations , the p roots obtained, 
2^i? y& *> %>? permute themselves cyclically when the variable x 
describes a loop around the origin We say that they form a cyclic 
system ofp roots If p = n, the n roots form a single cyclic system 
If p is less than n, we shall repeat the reasoning, starting with one 
of the remaining n p roots and so on. It is clear that if we con- 
tinue in this way we shall end by exhausting all the roots, and we 
can state the following proposition. The n roots of the equation 
F(x, y) = 0, which are zero for x = 0, form one or several cyclic 
systems in the neighborhood of the origin 

To render the statement perfectly general, it is sufficient to agree 
that a cyclic system can be composed of a single root ; that root is 
then a single-valued function in the neighborhood of the origin 

The roots of the same cyclic system can be represented by a unique 
development. Let y v y z , -, y p be the p roots of a cyclic system , let 
us put x = x*. Each of these roots becomes an analytic function 
of x' for all values of x ! other than y} = , on the other hand, when 
x 1 describes a loop around x r = 0, the point x describes p succes- 
sive loops m the same sense around the origin. Each of the roots 
2/i? y 2 > '> VP returns then to its initial value , they are single-valued 
functions in the neighborhood of the origin Since these roots ap- 
proach zero when x 1 approaches zero, the origin x' = cannot be 



V, 99] IMPLICIT FUNCTIONS 239 

otliei than an ordinary point, and one of these roots is represented 
by a development of the form 

(29) 2/ = X + *X 2 + '* +<v"-f..., 
or, replacing x* by x l/p , 

i / i\ 2 / iV 

(30) y = ^a* + a^ap; + + a m \x) + . 

We may now say that the development (30) represents all the roots 
of the same cyclic system, provided that we give to x lfp all of its 
p determinations For, let us suppose that, taking for the radical V& 
one of its determinations, we have the development of the loot y r 
If the variable x describes a loop around the origin in the positive 
sense, y^ changes into y^ and x l/p is multiplied by e 2lrt/ * It will be 
seen, similarly, that we shall obtain y q by replacing x l/p by x l/p e zq1ri/p 
in the equality (30) This unique development for the system shows 
up clearly the cyclic permutation of the^ roots It would now lemain 
to show how we could separate the n roots of the equation F(x, y) = 
into cyclic systems and calculate the coefficients a t of the develop- 
ments (30) We have already considered the case where the point 
x = y = is a double point (I, 199, 2d ed ) We shall now treat 
another particular case 

If f 01 x = y = the derivative dF/dx is not zero, the develop- 
ment of F(x, y) contains a term of the first degree in x, and we have 

(31) F(x,y) = 4x+Bf + , (AB*0) 

where the terms not written are divisible by one of the factors cc 2 , vy, 
y n+l Let us consider y for a moment as the independent vanable; 
the equation F(x, y) = has a single root approaching zero with y, 
and that root is analytic in the neighborhood of the origin The 
development which we have already seen how to calculate (I, 35, 
193, 2d ed , 20, 187, 1st ed.) runs as follows 

(32) s = 0"(*o + iy+ ) (*<>) 
Extracting the wth root of the two sides, we find 



(33) af 

For y = the auxiliary equation u n == a Q + a^j + has n dis- 
tinct roots, each of which is developable in a power series according 
to powers of y. Since these n roots are deducible from one of them 
by multiplying it by the successive powers of &* m/n , we can take for 
~\/a, Q + a~y + in the equality (33) any one of these roots, subject 
to the condition of assigning successively to x l/n its n determinations 



240 SEVERAL VARIABLES [V, 99 

We can therefore write the equation (33) in the form 

a = i 1 y + a i y a + , ft *> 0) 

and from this ve deiive, conversely, a development of y in poweis 



(34) ^ = ^ + 

This development, if we give successively to # Vn its n values, 
represents the n roots which appioach zero with x These n roots 
form, then, a single cyclic system 

Foi a study of the general case we lefei the reader to treatises 
devoted to the theory of algebraic functions * 

100. Algebraic functions Up to the present time the implicit func- 
tions most carefully studied aie the algebraic functions, defined by 
an equation F(x : y) = 0, in which the left-hand side is an irreducible 
polynomial m x and y A polynomial is said to be irreducible when 
it is not possible to find two other polynomials of lower degiee, F^x, y) 
and F 2 (#, y), such that we have identically 

F(x 9 y) = F&, y) X Fx, y). 

If the polynomial F(x, y) were equal to a product of that kind, it is 
clear that the equation F(x, y) = could be replaced by two distinct 
equations Ffo, y) = 0, F 2 (x, y) = 
Let, then, 

(35) F(x,y) = 4>,(xW+4> l (z)p-' i + + *,-i(B)y+* i (aO=0 

be the proposed equation of degree n in ?/, where < , $ v , <f> n are 
polynomials in % Eliminating ?/ between the two relations F = 0, 
dF/dy = 0, we obtain a polynomial A(OJ) for the resultant, which can- 
not be identically zero, since F(x, y) is supposed to be irreducible. 
Let us rnaik in the plane the points a v a,, , (%, which represent 
the roots of the equation A(x)= 0, and the points p v yS 2 , - , p h , 
which represent the roots of < (&)= Some of the points <r z may 
also be among the roots of < (&)= For a point a different from 
the points a l3 fa the equation F(a,y*)= has n distinct and finite 
roots, # 1? 5 2 , * - -, b n In the neighborhood of the point a the equation 
(35) has therefore n analytic roots which approach b v & 2 , , b n 
respectively when x approaches a. Let a % be a root of the equation 



* See also tlie noted memoir of Puaseux on algebraic functions (Journal de MatM- 
mattques, Vol XV, 1850) 



V,100] ALGEBRAIC FUNCTIONS 241 

A (#) = The equation F(a iy y) = has a certain numbei of equal 
roots , let us suppose, for example, that it has p roots equal to I 
The p roots which appioach Z> when x appi caches tr, group themselves 
into a ceitam numbei of cyclic systems, and the roots of the same 
cyclic system are represented by a development 111 series arranged 
accoiding to fiactional powers of x # t If the value a t does not 
cause <j5> (&) to vanish, all the roots of the equation (35) in the neigh- 
boihood of the point a l group themselves into a certain number of 
cyclic systems, some of which may contain only one root For a point 
fa which makes < (#) zero, some of the loots of the equation (35) 
become infinite , in order to study these roots, we put y == !//, and 
we are led to study the roots of the equation 



which become zeio for x = fa These roots group themselves again 
into a certain number of cyclic systems, the loots of the same system 
being repiesented by a development in senes of the form 

(36) y' = o(a - &) * + a m+l (x - ft)^ + > K * 0) 

The coiresponding roots of the equation in y will be given by the 
development 

(37) y = (x - fl)~*[rti. + *(* - ft) >+ 



which can be arranged in increasing powers of (r J3,) l/p , but there 
will be at first a finit e number of terms with negative exponents. 

To study the values of y f 01 the infinite values of x, we put x = l/# f , 
and we are led to study the roots of an equation of the same form in 
the neighborhood of the origin To sum up, in the neighborhood of 
any point x = a the n roots of the equation (35) are represented by 
a certain number of senes arranged according to increasing powers 
of x a or of (x a) l/p , containing perhaps a finite numbei of terms 
with negative exponents, and this statement applies also to infinite 
values of x by replacing x oc by 1/cc. 

It is to be observed that the fractional powers or the negative ex- 
ponents present themselves only for the exceptional points The 
only singular points of the roots of the equation aie therefore the 
critical points around which some of these roots permute themselves 
cyclically, and the poles where some of these roots become infinite ; 
moreover, a point may be at the same time a pole and a critical 
point These two kinds of singular points are often called algebraic 
singular po^nts. 



242 SEVERAL VARIABLES [V, 100 

We have so far studied the roots of the proposed equation only in 
the neighborhood of a fixed point Suppose now that we ]0in two 
points x = a, x = b, for which the equation (35) has n distinct and 
finite roots, by a path AB not passing through any smgulai point of 
the equation Let y^ be a root of the equation F(a, y) = , the root 
y =/(x), which reduces to y l for x = a, is represented in the neigh- 
borhood of the point a, by a power-series development P (x a) 
We can propose to ourselves the problem of finding its analytic ex- 
tension by causing the variable to describe the path AB This is a 
particular case of the general problem, and we know in advance that 
we shall arrive at the point B with a final value which will be a 
root of the equation F(b, y) = ( 86) We shall surely arrive at 
the point 1} at the end of a finite number of operations , in fact, the 
radii of the circles of convergence of the series representing the 
different roots of the equation F(x, y) = 0, having their centers at 
different points of the path AB } have a lower limit* 8 >0, since this 
path does not contain any critical points ; and it is clear that we 
could always take the radii of the different circles which we use for 
the analytic extension at least equal to 8 

Among all the paths joining the points A and B we can always 
find one leading from the root y^ to any given one of the roots of 
the equation F(l, y) = as the final value The proof of this can be 
made to depend on the following proposition If an analytic func- 
tion & of the variable x has only p distinct values for each value ofx, 
and if it has in the whole plane (including the point at infinity) only 
algebraic singular points, the p determinations of z are roots of an 
equation of degree p whose coefficients are rational functions of x 
Let s 1? z 2 , - , z p be the p determinations of z , when the variable x 
describes a closed curve, these p values z v z^ , # p can only change 
into each other. The symmetric function u k = z\ 4- 4, + + > 
where & is a positive integer, is therefore single-valued Moreover, 
that function can have only polar singularities, for in the neigh- 
borhood of any point in the finite plane x = a the developments 
ot z v s 2 , - -, p have only a finite number of terms with negative 
exponents. The same thing is therefore true of the development of %. 
Also, the function % being single-valued, its development cannot con- 
tain fractional powers The point a is therefore a pole or an ordinary 
point for %., and similarly for the point at infinity. The function u k 



* To prove this rigorously it suffices to make use of a form of reasoning analogous 
to that of S 84 



V, 101] ALGEBRAIC FUNCTIONS 243 

is therefore a rational function of x, whatever may be the integer 
70, consequently the same thing is true of the simple symmetric 
functions, such as S,, S^,*?*, > which proves the theoiem stated 
Having shown this, let us now suppose that 111 going from the 
point a to any other point x of the plane by all possible paths we 
can obtain as final values only p of the roots of the equation 

F(x,y)=Q, CP<) 

These p roots can evidently only be permuted among themselves 
when the variable x describes a closed boundaiy, and they possess 
all the properties of the p branches 19 & 2) , & p of the analytic 
function which we have just studied. We conclude from this that 
T/ I? y^ , y p would be roots of an equation of degree p, F^x, y) = 0, 
with rational coefficients The equation F(x f y) would have, 
then, all the roots of the equation Ffo, y) = 0, whatever x may be, 
and the polynomial F(x, y) would not be irreducible, contrary to 
hypothesis If we place no restriction upon the path followed by 
the variable x, the n roots of the equation (35) must then be regarded 
as the distinct branches of a single analytic function, as we have 
already remarked in the case of some simple examples (6). 

Let us suppose that from each of the critical points we make an 
infinite cut in the plane in such a way that these cuts do not cross 
each other. If the path followed by x is required not to cross any 
of these cuts, the n roots are single-valued functions in the whole 
plane, for two paths having the same extremities will be transform- 
able one into the other by a continuous deformation without passing 
over any critical point ( 85) In order to follow the variation of a 
root along any path, we need only know the law of the permutation 
of these roots when the variable describes a loop around each of the 
critical points 

Note The study of algebraic functions is made relatively easy by the fact 
that we can determine a priori by algebraic computation the singular points of 
these functions This is no longer true in general of implicit functions that are 
not algebraic, which may have transcendental singular points As an example, 
the implicit function y (x) , defined by the equation & # 1 = 0, has no algebraic 
critical point, but it has the transcendental singular point x = 1. 

101 . Abelian integrals. Every integral I=>fR (a?, y) dx, where E (as, y) 
is a rational function of x and y, and where y is an algebraic func- 
tion defined by the equation F(x, y) = 0, is called an Abelian integral 
attached to that c^trve. To complete the determination of that inte- 
gral, it is necessary to assign a lower limit X Q and the corresponding 



244 SEVERAL VARIABLES [V, 101 

value ?/ chosen among the roots of the equation F(x Q , */)= We 
shall now state some of the most important geneial properties of such 
integials When we go from the point x to any point x by all the 
possible paths, all the values of the mtegial I are included in one 
of the formulae 



(38) JT:=7 Ji + m 1 ai 1 + m a tt a + + > r <D r , (& = 1>2> - ,72,) 

where 7 13 7 2 , , I n are the values of the integral which correspond 
to certain definite paths, m v m 2 , - , m r are arbitrary integers, and 
o> 1? a>>, , o>, are penods These periods are of two kinds , one kind 
results fiom loops described about the poles of the function It (a, y) , 
these are the polar periods The others come from closed paths 
surrounding several critical points, called cycles, these are called 
cyclic periods. The numbei of the distinct cyclic periods depends 
only on the algebraic relation considered, F(r, y) , it is equal 
to 2p, where p denotes the deficiency of the cuive ( 82) On the 
other hand, there may be any number of polar periods From the 
point of view of the singularities three classes of Abelian integrals 
aie distinguished Those which remain finite in the neighborhood 
of every value of x are called the first kind-, if their absolute value 
becomes infinite, it can only happen thiough the addition of an 
infinite number of periods The integrals of the second kind are 
those which have a single pole, and the integrals of the third Jcmd 
have two logarithmic singular points Every Abelian integral is a 
sum of integrals of the three kinds, and the number of distinct 
integials of the first kind is equal to the deficiency 

The study of these integrals is made very easy by the aid of plane 
surfaces composed of several sheets, called Riemann surfaces We 
shall not have occasion to consider them here We shall only give, 
on account of its thoroughly elementary character, the demonstrar 
tion of a fundamental theorem, discovered by Abel 

102. Abel's theorem. In order to state the results more easily, let us 
consider the plane curve C represented by the equation F(x, y) = 0, 
and let $ (x, y) be the equation of another plane algebraic curve C r 
These two curves have N points in common, (x v y^), (# 2 , y^, , 
( x m VN)) the number N being equal to the product of the degrees 
of the two curves Let R(x, y) be a rational function, and let us 
consider the following sum 



(39) jW JZ(x,y)fc, 



V,102] ALGEBRAIC FUNCTIONS 245 

where 

/&*? 

R(x,y}dx 



/&*?) 

I 

/ (*o yd 



denotes the Abelian integral taken from the fixed point X Q to a point x 
along a path which leads y from the initial value y to the final value y^ 
the initial value ?/ of y being the same for all these integrals It is 
clear that the sum / is determined except f 01 a period, since this is 
the case with each of the integrals Suppose, now, that some of the 
coefficients, a v a^ , a l} of the polynomial <(#, y) are variable 
When these coefficients vary continuously, the points # t themselves 
vary continuously, and if none of these points pass through a point 
of discontinuity of the integral fR (x, y) dx } the sum J itself varies 
continuously, provided that we follow the continuous variation of 
each of the integrals contained in it along the entire path described 
by the corresponding upper limit. The sum / is therefore a function 
of the parameters a v a 2 , , a ly whose analytic form we shall now 
investigate 

Let us denote in general by 87 the total differential of any func- 
tion V with respect to the variables & 1? & 2 , , a L : 



By the expression (39) we have 

From the two relations F(x l7 y^) = 0, $(x t , 2/ t ) = we derive 



and consequently 8# t = V(x % , 2/ t )8$ t , where ^(x^y^) is a rational 
function of x l} y^ a v a^ , a L , and where <E> t is put for 
We have, then, ^ N 



The coefficient of 8^ on the right is a rational symmetric function 
of the coordinates of the N points (x l} y^) common to the two curves 
C, C" The theory of elimination proves that this function is a 
rational function of the coefficients of the two polynomials F(x, y) 
and $(#, y) y and consequently a rational function of a v a z , , a k 
Evidently the same thing is true of the coefficients of 8& 2 , , 8^, 



246 SEVERAL VARIABLES [V,102 

and I will be obtained by the integration of a total differential 



/ = f 

where TT I? IT,, , IT L aie lational functions of a v c& 2 , , % Now 
the integration cannot introduce any other tianscendentals than 
logarithms. The sum I is therefore equal to a, rational function of 
the coefficients a lt a v , a l9 plus a sum of logarithms of rational 
functions of the same coefficients, each of these logarithms being 
'multiplied bi/ a constant factor This is the statement of Abel's 
theorem in its most general form In geometiic language we can 
also say that the sum of the values of any Abelian integral, taken 
from a common origin to the N points of intersection of the given 
curve with a variable curve of degree m, $(x, y)= 0, is equal to a 
rational function of the coefficients of *(x, y),plu* <*> sum of a finite 
number of logarithms of rational functions of the same coefficients, 
each logarithm being multiplied by a constant factor 

The second statement appears at first sight the more striking, 
but in applications we must always keep in mind the analytic state- 
ment in the evaluation of the continuous variation of the sum / 
which corresponds to a continuous vanation of the parameters 
a lt a a , , a L . The theorem has a precise meaning only if we take 
into account the paths described by the N points x l9 a? a , , X N on 
the plane of the variable x. 

The statement becomes of a remarkable simplicity when the 
integral is of the first kind In fact, if TT I? 7T 2 , , TT^ were not 
identically zero, it would be possible to find a system of values 
a^ = a{, , a k = a{ for which I would become infinite. Let (arj, yj), 
. , (tfy, 1/tf) be the points of intersection of the curve C with the 
curve C' which correspond to the values a[ , , a' K of the parameters 
The integral ^ 

I 

/( 



would become infinite when the upper limit approaches one of the 
points (o, 2/0* which is impossible if the integral is of the first kind 
Therefore we have S/ = 0, and, when a v a^ , % vary continuously, 
/ remains constant , Abel's theorem can then be stated as follows . 

Given a fixed curve C and a variable curve C 1 of degree m, the sum 
of the increments of an Abelian integral of the first kind attached to 
the curve C along the eontmuous curves described by the points of 
intersection of C with C 1 is equal to zero. 



V,loa] ALGEBRAIC FUtfCTIOSTS 247 

Note We suppose that the degree of the curve C* remains con- 
stant and equal to m If for certain particular values of the coeffi- 
cients a v & 2 , , a k that degree weie lowered, some of the points of 
intei sections of C with C' should be regarded as thrown off to 
infinity, and it would be necessary to take account of this in the 
application of the theorem We mention also the almost evident fact 
that if some of the points of intersection of C with C" are fixed, it 
is unnecessaiy to include the corresponding integrals in the sum L 

103 Application to hyperelliptic integrals The applications of 
AbePs theorem to Analysis and to Geometry are extremely numer- 
ous and important We shall calculate SI explicitly in the case of 
hypeielliptic integrals. 

Let us consider the algebraic relations 



(40) ^ = (aO= 
where the polynomial R(x) is prime to its derivative We shall 
suppose that A Q may be zero, but that A Q and A l may not be zero at 
the same time, so that R (x) is of degree 2p + 1 or of degree 2^ + 2 
Let Q(x) be any polynomial of degree q We shall take for the 
initial value X Q a value of x which does not make R (x) vanish, and 
f 01 y a root of the equation y 1 = R (a? ). We shall put 




where the integral is taken along a path going from x to x, and 
where y denotes the final value of the radical V,R(a;) when we start 
from x with the value y Iw order to study the system of points 
of intersection of the curve C represented by the equation (40) with 
another algebraic curve C', we may evidently replace in the equation 
of the latter curve an even power of y, such as f r > by [-K()] r , and 
an odd power y* r+i by y[^()T These substitutions having been 
made, the equation obtained will now contain y only to the first 
degree, and we may suppose the equation of the curve C' of the form 



(41) 

where /(a?) and < (x) are two polynomials prune to each other, of 
degrees X and /* respectively, some of the coefficients of which we 
shall suppose to be variable The abscissas of the points of intersec- 
tion of the two curves C and C' are roots of the equation 

(42) j( 



248 SEVERAL VARIABLES [V, 103 

of degree N. For special systems of values of the variable coefficients 
m tlie two polynomials f(x) and < (x) the degiee of the equation may 
turn out to be less than A 7 , some of the points of intersection are 
then thiown off to infinity, but the corresponding integrals must 
be included in the sum which we aie about to study To each root 
x l of the equation (42) corresponds a completely determined value 
of y given by y=/(i)/^(^i) ^et us now consider the sum 



We have 



for the final value of the radical at the point x t must be equal to 
?/ 1? that is, to f(x t )/<j> (a 1 ,) On the other hand, from the equation 
^(aj t )= we derive 

</,'(*,) te. + 2 * (as,) < (*,) 8*. - 2/fo) 8/, = 0, 
and therefore 




or, making use of the equation (42), 



Let us calculate, for example, the coefficient of Sa k in 87, where a fc is 
the coefficient, supposed variable, of x 1 in the polynomial f(x) The 
term 8% does not appear in 8$,, and it is multiplied by x% in 8/ t The 
desired coefficient of 8a k is therefore equal to 

a * _ 



where ?r(a;) = Q (a;) <jf> (#) aj x . The preceding sum must be extended to 
all the roots of the equation i/r(x) = , it is a rational and symmetric 
function of these roots, and therefore a rational function of the coeffi- 
cients of the two polynomials f(x) and < (x). The calculation of 
this sum can be facilitated by noticing that STT^)/^'^) is equal to 
the sum of the residues of the rational function 7r(x)/iff(x) relative 
to the N poles in the finite plane x v j 2 , ,X N By a general theo- 
rem that sum is also equal to the residue at the point at infinity 
with its sign changed ( 52) It will be possible, then, to obtain the 
coefficient of 8a L by a simple division. 



V, 103] ALGEBRAIC FUNCTIONS 249 

It is easy to prove that this coefficient is zero if the integral 
v(cc, y) is of the first kind We have by supposition q ^ p 1 , the 
degree of TT(CC) is q -h p + k, and we have 



Let us find the degree of ifr(x). If there is no cancellation between 
the terms of highest degree in R(x) <^(x) and m/ 3 (o;), we have 

2\ 

whence 

and, a fortiori, 

If there were a cancellation between these two terms, we should have 

A, =: 



but since the term a k x^^ k has no term with which to cancel out, we 
should have \ + & ^ N, from which the same inequality as before 
lesults It follows that we always have 



The residue of the rational function 7r(x)/ij/(x) with respect to the 
point at infinity is therefore zero, for the development will begin 
with a term in 1/x 2 or of higher degree. It will be seen similarly 
that the coefficient of Bb h in 87, b h being one of the variable coefficients 
of the polynomial <(#), is zero if the polynomial Q(x) is of degree 
p 1 or of lower degree This result is completely in accord with 
the general theorem. 

Let us take, for example, < (x) = 1 3 and let us put 



a p x* 



wheie a , a l9 , a p are p + 1 variable coefficients The two curves 



cut each other in 2p + 1 variable points, and the sum of the values 
of the integral v (z, y), taken from an initial point to these 2 p + 1 
points of intersection, is an algebraic-logarithmw function of the 
coefficients a , a l9 , a p Now we can dispose of these p -f 1 coeffi- 
cients in such a way that p -f 1 of the points of iDtersection are any 
previously assigned points of the curve ^=^R(x)^ and the coordi- 
nates of the p remaining points will be algebraic functions of the 
coordinates of the p + 1 given points. 



250 SEVERAL VARIABLES [V, 103 

The sum of the^> + 1 integrals 



taken from a common initial point to p 4- 1 arbitrary points, is 
therefore equal to the sum of p integrals whose limits are algebraic 
functions of the coordinates 



plus certain algebiaic-logarithmic expressions It is clear that by 
successive reductions the proposition can be extended to the sum 
of m integrals, where m, is any integer gi eater thanj? In particular, 
the sum of any number of integrals of the first kind can be i educed 
to the sum of only p integrals. This propeity, which applies to the 
most general Abelian integrals of the first kind, constitutes the 
addition theorem for these integrals. 

In the case of elliptic integrals of the first kind, Abel's theorem leads pre- 
cisely to the addition formula for the function p(u) Let us consider a cubic in 
the normal form 



and let Mjfa, y^ 3f 2 (x 2 , y a ), ^ S (x 3 , 2/ 8 ) be the points of intersection of that 
cubic with a straight line D By the general theorem the sum 




is equal to a period, for the three points M v Jf s , 3f 3 are carried off to infinity 
when the straight line D goes off itself to infinity Now if we employ the 
parametric representation x =p(w), y = P'(M) for the cubic, the parameter u is 
precisely equal to the integral 



I - ._ 



and the preceding formula says that the sum of the arguments M I? u 2 , w 8 , which 
correspond to the three points M v Jf 2 , Jkf s , is equal to a period We have seen 
above how that relation is equivalent to the addition formula for the function 
p(u)(80). 

104. Extension of Lagrange's formula The general theorem on the implicit 
functions defined by a simultaneous system of equations (I, 194, 2d ed , 
188, 1st ed) extends also to complex variables, provided that we retain 
the other hypotheses of the theorem. Let us consider, for example, the two 
simultaneous equations 



(44) P(a,y) = fc-a-a/(x,y) = 0, Q(ai, y) = y-6- p<l>(x, y) = 0, 

where x and y are complex variables, and where /(<c, y) and <f> (x, y) are ana- 
lytic functions of these two variables in the neighborhood of the system of 



V,104] ALGEBRAIC FUNCTIONS 251 

values x = a, y = b For a = 0, /3 = these equations (44) have the system of 
solutions x = a, y = 6, and the deteiminant D(P, Q)/D(x, y) leduces to unity 
Theiefoie, by the general theorem, the system of equations (44) has one and 
only one system of roots appi caching a and 6 lespectively when a and /3 approach 
zero, and these loots are analytic functions oi a and p Laplace was the first 
to extend Lagrange's formula ( 51) to this system ot equations 

Let us suppose for defimteness that with the points a and & as centers we 
describe two circles C and <7' in the planes of the variables x and y respectively, 
with ladn r and r 7 so small that the two functions /(x, y) and <f> (x, y) shall be 
analytic when the vanables x and y remain within 01 on the boundaries of 
these two ciicles (7, C' Let M and M' be the maximum values of |/(x, y) \ and 
of |0(x, y)|, lespectively, in this region We shall suppose fuither that the 
constants a and /5 satisfy the conditions M \ a \ < r, If' | /3 1 < r 7 

Let us now give to x any value within 01 on the boundary of the ciicle (7, 
the equation Q (x, y) = is satisfied by a single value of y in the interior of the 
circle <7', for the angle of y & (x, y) increases by 2 TT when y describes 
the circle C' in the positive sense ( 49) That root is an analytic function 
y l = ^ (x) of x in the circle (7 If we replace y in P (x, ?/) by that root y t , the 
resulting equation x a ar/(x, y a ) = has one and only one root in the inte- 
1101 of (7, for the reason given a moment ago 

Let x = be that root, and let t\ be the corresponding value of y, t\ = ^ () 
The object of the generalized Lagrange formula is to develop in powers of a, 
and /3 eveiy function Ffa 17) which is analytic m the region just defined 

For this purpose let us consider the double integral 

MK\ r C A* C fffe y)&y 

(QtO) JL s= I UfX I 

Since x is a point on the circumference of (7, P(x, y) cannot vanish for any 
value of y within C', for the angle of x a a/(x, y) returns necessarily to 
its initial value when y describes <7 X , x being a fixed point of C The only pole 
of the function under the integral sign, considered as a function of the single 
variable y, is, then, the point yy v given by the root of the equation Q (x, y) = 0, 
which corresponds to the value of x on the boundary C, and we have, after a 
first integration, 

C F(x^y)dy >. ff(x, yi) 

Jcc")P(a, v) Qfa y) " ^ p/ v x /a? 
v ^ i; \dy 

The right-hand side, if we suppose y^ replaced by the analytic function $ (x) 
defined above, has in turn a single pole of the first order in the interior of C, 
the point x = ft to which corresponds the value y l = % and the corresponding 
residue is easily shown to be 



The double integral I has therefore for its value 

I=-4* rj ( p f yi '* 



252 SEVEEAL VARIABLES [V, 104 

On the other hand, we can develop l/PQ m a uniformly conveigent series 



b-p<p) <*-i (x - 
which gives us I =SJ r mn a/3, where 

r = r fa r -Ffe y) E/te y)] m E* to y)?^ 

mn J<C) J7) (aj-a)+i(tf-&)+ 1 

This integral has already heen calculated ( 94), and we have found that it is 
47T 2 d 



Equating the two values of I, we obtain the desired result, which presents an 
evident analogy with the formula (50) of 51 



"We could also obtain a second result analogous to (51), of 51, by putting 



but the coefficients in this case are not so simple as in the case of one variable 

EXERCISES 

1, Every algebraic curve C n of degree n and of deficiency jp can be earned 
over by a birational transformation into a curve of degree p + 2 

(Proceed as in 82, cutting the given curve by a net of curves <7 n _ 2 , passing 
through n (n 1)/2 3 points of (7 n , among which are the (n 1) (n 2)/2 p 
double points, and put 



the equation of the net being ^ x (aj, y) + X0 2 (a>, y) + ^0 8 (aj, y) = 0.) 

2 Deduce from the preceding exercise that the coordinates of a point of a 
curve of deficiency 2 can be expiessed as rational functions of a parameter t 
and of the square root of a polynomial R(t) of the fifth or of the sixth degiee, 
prime to its derivative 

(The reader may begin by showing that the curve corresponds point by point 
to a curve of the fourth degree having a double point ) 

3* Let y = a^x + <x z x 2 + - be the development in power series of an alge- 
braic function, a root of an equation F(x, y) = 0, where F(x, y) is a polynomial 
with integral coefficients and where the point with coordinates x = 0, y = is a 
simple point of the curve represented by Ffay) = Q All the coefficients e^, <ar a 
are fractions, and it suffices to change x to Jfo, K being a suitably chosen integer, 
in order that all these coefficients become integers, [EISENSTEIN ] 

(It will be noticed that a transformation of the form x = Wx', y ky' suffices 
to make the coefficient of y" on the left-hand side of the new relation equal to 
one, all the other coefficients being integers.) 



INDEX 



[Titles m italic are proper names, immbeis in italic are page numbers , and num- 
bers in roman type are paragraph numbers ] 



Abel : 19, f tn , 170, 76 , 180, 78 , 198, 
82 , 244, 101 

Abelian mtegials : see Integrals 

Abel's theorem : 244, 102 

Addition formulae : 27, 12; for elliptic 
functions : 166, 74 , $50, 103 

Adjoint curves : 191, 82 

Affixe : 4, f tn 

Algebraic equations : see Equations 

Algebraic functions : see Functions 

Algebraic singular points : see Singular 
points 

Analytic extension : 196, 83 , 199, 84 ; 
functions of two variables : 31, 97 

Analytic functions: 7, 3 , 11, 4; ana- 
lytic extension: 196, 83, 199, 84 J 
derivative of : 9, 3, 48, 9, 77, 33; 
elements of : 198, 83 ; new definition 
of : 199, 84 ; series of : 86, 39 ; zeros 
of : 88, 40 , see also Cauchy's theo- 
rems, Functions, Integral functions, 
Single-valued analytic functions 

Analytic functions of several varia- 
bles: 218, 91; analytic extension 
of: 281, 97; Cauchy's theorems: 
225, 94, 227, 95; Lagrange's for- 
mula : 250, 104 ; Taylor's formula : 
222, 92 , 226, 94 ; singularities of : 
232, 97 

Anchor ring : 54, ex 3 

Appell : 84, 38 , 217, ex. 3 

Associated circles of convergence : 
220, 92 

Associated integral functions: 218, 
ex 7 

Bertrand : 58, ex 22 
Bicircular quartics : 198, ex. 
Binomial formula : 40, 18 



Birational transformations : 192, 82 , 

252, ex 1 

Blumenthal : 188, f tn. 
Borel: 180, ftn , 138, ftn , .?, ex. 3 
Bouquet . see Btiot and Bouquet 
Branch point: see Critical points 
Branches of a function : 15, 6 , 29, 13 
Bnot and Bouquet : 126, ex 27 , 195, 

ex 11 

Burman: 126, ex 26 
Burman's series : 126, ex 26 



7, 2 , 0, ftn , j?0, 3 ; 60, 25 , 
7jf, ftn , 74, 32 , 7*, 34 ; 82, ftn , 
106, 51 , ^4 5 53 , 127, 57 , .Z30, 63 , 
200, ftn , 216, 90 , &8, 93 , 886, 94 , 
jft*7, 95 , 838, 98 

Cauchy-Laurent series : 81, 35 

Cauchy's integral* 75, 33; funda- 
mental formula: 76, 33, 7, 95; 
fundamental theorem : 283, 98 , in- 
tegral theorems: 75, 33; method, 
Mittag-Leffler's theorem: 189, 63; 
theorem: 66, 28, 7-?, ftn , 74, 32, 
75, 33 , 78, 34 , 816, 90; theorem for 
double integrals : 888, 93 , 886, 94 

Cauchy-Taylor series : 79, 35 

Change of variables, in integrals : 62, 26 

Circle of convergence : 18, 8 , 808, 
84, #00, 87, 818, 88; associated 
circles of convergence: 880, 92; 
singular points on : 808, 84 and ftn 

Circular transformation : 45, 19 , 57, 
ex 13 

Class of an integral function : 182, 58 

Clebschi 186, ftn. 

Complex quantity : 8, 1 

Complex variable: 6, 2; analytic func- 
tion of a : 9, 3 ; function of a : 6, 2 



253 



254 



INDEX 



Conf ormal maps : see Maps 

Confoimal repiesentation : 4%, 19 , 45, 
20 , 48, 20 , 52, 23 , see also Projec- 
tion and Transformations 

Conf ormal transformations: seeTians- 
formations 

Conjugate imaginanes: 4, I 

Conjugate isothermal systems : 54, 24 

Connected region . 11, 4 

Continuity, of functions : 6, 2; of 
power series 7, 2 , 56, ex 7 

Continuous functions : see Functions 

Convergence, circle of: see Circle of 
convex gence 

Convergence, unifoim: of infinite 
products: 82, 10 , 129, 57; of inte- 
grals : 289,M ; of series . 7, 2 , 88, 39 

Cousin : 232, f tn 

Critical points : 15, 6 , 0, 13 , 837, 
99; logarithmic: 88, 14, 113, 53 

Cubics : see Curves 

Curves, adjoint: 191, 82; bicircular 
quartics: 193, ex ; conjugate iso- 
thermal systems : 54, 24 ; deficiency 
of : 172, 77 f 191, 82 , 25%, exs 1 and 
2 , 244, 101 ; of deficiency one : 178, 
77 ; double points : 184, 80, 191, 82 ; 
loxodromic: #, ex 1, parametnc 
representation of curves of defi- 
ciency one : 187, 81 , 191, 82 , 19S, 
ex ; parametric representation of 
plane cubics . 180, 78 , 184, 80 , 187, 
81; points of inflection: 186, 80; 
quartics : 187, 81 ; unicursal : 191, 
82, see also Abel's theorem and 
Rhumb lines 

Cuts : 08, 87 

Cycles: 244, 101 

Cyclic periods : 244, 101 

Cyclic system of roots : 238, 99 

D'Alemberti 104, Note 

D' Alembert's theorem : 104, Note 

Darboux : 64, 27 

Darboux's formula, law of the mean : 

64,27 

Deficiency : see Curves, deficiency of 
Definite integrals: 60, 25, 72, 31, 

97, 46; differentiation of: 77, 33; 



27, 95; evaluation of: 96, 45; 
FresnePs : 100, 46 ; F function : 
100, 47 , 249, 96 ; law of the mean : 
64, 27; penods of. 118, 53, ^4, 
Note, see also Integials 

De Moivre : 6, 1 

De Moivre's formula, 6, 1 

Denvative, of analytic functions: 9, 
3 , 42, 19 , 77, 33 ; of integrals . 77, 
33, 287, 95, of power senes. 19, 
8; of senes of analytic functions: 
8S,B9 

Dominant function : 56, ex 7 , 81, 

QK 00v 04. 
oo , &&/, y* 

Dominant senes . &Z, 9 , 157, 69 
Double integrals : 888, 93 ; Cauchy's 

theorems : 882, 93 , 285, 94 
Double points : 184, 80 , ^PJf, 82 
Double series : 81, 9 ; circles of con- 
vergence: 830, 92; Tayloi's for- 
mula : 882, 92 , S86, 94 
Doubly periodic functions' 145, 65, 
149, 67 , see also Elliptic functions 

Eisenstem : 252, ex 3 

Elements of analytic functions : 198, 
83 

Elliptic functions: 145, 65, 150, 08; 
addition formulae: 1#, 74; alge- 
braic relation between elliptic func- 
tions with the same periods: 15S 
68 ; application to cubics : 180, 78 , 
184, 80; application to curves of 
deficiency one : 187, 81 , 191, 82 ; 
application to quartics: 187, 81; 
even and odd : 154, 68 ; expansions 
for : 154, 69 , general expression for: 
168, 73; Hermite's formula: 165, 
73 , 168, 75 , 195, ex 9 ; integration 
of : 168, 75 ; invariants of : 158, 70 , 
172, 77 , 182, 79 ; order of : 150, 68 ; 
p(w) : 154, 69; p(u) defined by in- 
variants : 182, 79 ; periods of : 158, 
68 ; 172, 77 , 184, 79 ; poles of : 150, 
68 , 154, 68 ; relation between p(u) 
and p' (it) : 158, 70 ; residues of : 
151, 68; <r(u) : Jft*, 72; 0,(u) : 170, 
76; r(u): 150, 71; zeros 6f: jtf*, 
68 , 154, 68 , jtf0, 70 



INDEX 



255 



Elliptic integrals, of the fiist kind: 
120, 56 , 174, 78 , 250, 103 ; the in- 
verse function : 174, 78 ; periods of : 
120, 56 

Elliptic transformation : $7, ex 15 

Equations: 283, 98; algebraic: 240, 
100 , cyclic system of roots of : 238, 
99, 241, 100; D'Alembert's theo- 
rem: 104, Note; Keplei's: 109, 
ex , 126, ex 27 ; Laplace's : 10, 3 , 
54, 24 , 55, Note ; theory of equa- 
tions : 10$, 49 , see also Implicit 
functions, Lagrange's formula, and 
Weierstrass's theorem 

Essentially singular point : 91, 42 ; at 
infinity : 1 10, 52 ; isolated : 91, 42 ; 
see also Laurent's senes 

Eul& : 27, 12 , 58, exs 20 and 22 , 
96, 45, 124, ex 14, 143, ftn., 830, 
96 

Euler's constant: 230, 96; formula: 
58, ex 22, 96, 45, 124, ex 14, 
formulae : 27, 12 

Evaluation of definite integrals : see 
Definite integrals 

Even functions : 153, Notes 

Expansions in infinite products: 
194, exs 2 and 3 ; of cos z : 194, 
ex 3; of T(z). 230, 96; of <r(u): 
162, 72 ; of sin cc :' 143, 64 , see also 
Functions, primary, and Infinite 
products 

Expansions in series : of ctn x : 143, 
64 ; of elliptic functions : 154, 69 ; 
of periodic functions: 145, 65; of 
loots of an equation: 888, 99; see 
also Series 

Exponential function : 23, 11 

Fourier: 170, 76 

Fredholm : 213, ftn 

Fuchs : 57, ex 15 

Fuchsian transformation : 57, ex 15 

Functions, algebraic : 233, 98 , 240, 
100; analytic: see Analytic func- 
tions and Analytic functions of sev- 
eral variables; analytic except for 
poles: 90, 41, 101, 48, 1S6, 61; 
branches of : 15, 6 , 29, 13 ; class 



of integial: 132, 58; of a complex 
variable: 6, 2; continuous: 6, 2; 
defined by differential equations: 
208, 86; dominant: 56, ex. 7, 81, 
35, 227, 94; doubly periodic: 145, 
65, .Z40, 67; elementary transcen- 
dental : 18, 8 ; elliptic : see Elliptic 
functions; even and odd: 153, 
Notes; exponential: 23, 11; Gamma: 
100, 47 , 229, 96; holomorphic: 11, 
ftn ; implicit : 233, 98 ; integral : 
see Integial functions and Integral 
transcendental functions; inveise, 
of the elliptic integral : 172, 77 ; in- 
verse sine: 114, 54; mveise trigo- 
nometric : 30, 14 ; irrational : 13, 
6; logarithms: 28, 13; meromor- 
phic: 90, ftn ; monodromic: 17, 
ftn ; monogenic : 9, ftn ; multiform : 
17, ftn ; multiple-valued : 17, 7 ; 
p(u) : 154, 69 ; periods of : 145, 65 , 
152, 68 , 178, 77 , 184, 79 ; primary 
(Weierstrass's) : 127, 57 ; pumitive : 
3, 15; rational: 12, 5, 33, 15; 
rational, of sm z and cos 2 . 85, 16 ; 
regular in a neighborhood . 0, 40 ; 
regular at a point : 88, 40 ; regular 
at the point at infinity : 109, 52 ; 
represented by definite integrals: 
227, 95; senes of analytic: 86, 39; 
<r (u) : 152, 72 ; single-valued : see 
Single-valued functions and Single- 
valued analytic functions; 0(w): 
170, 76; trigonometric: 26, 12; 
f (M) : 15P, 71 ; see ateo Expansions 

Fundamental formula of the integral 
calculus : 63, 26 , 72, 31 

Fundamental theorem of algebra: 
104, Note 

Gamma function : 1 00, 47 , 229, 96 
Gauss : 125, ex. 21 
Gauss's sums : 125, ex 21 
General linear transformation: 44, 

ex 2 

Geographic maps : see Maps 
Gouner : 126, ex 28 
Goursat: 208, ftn.; #.?, ftn 
Goursat's theorem : 69, 29 and ftn. 



256 



INDEX 



Hadamatd: 206 ', ftn , 212, 88, 2 '18, 
ex 8 

Hermite : 106, 51 , 100, ex , 165, 73 , 
Jf&S, 75 , 105, ex 9 , 215, 90 and 
ftn , 216, ftn , j?7, exs 4, 5, 6 , 
230, ftn 

Hermite's formula : 215, 90 ; for ellip- 
tic integrals: 165, 73 , 168, 75 , .Z05, 
ex 9 

Holomorphic functions : 11, ftn 

Hyperbolic tiansformations: 57, ex 
15 

Hypei elliptic integrals. 116, 55, #47, 
103; periods of : 116, 55 

Imagmaries, conjugate : 4, 1 

Imaginary quantity : 8, 1 

Implicit functions, Weierstrass's theo- 
rem : 233, 98 , see also Functions, 
inverse, and Lagrange's formula 

Independent periods, Jacobi's theo- 
rem : 147, 66 

Index of a quotient : 103, 49 

Infinite number, of singular points; 
134, 60, see also Mittag-Leffler's 
theorem 5 of zeros: 26, 11 , 93, 42 , 
128, 57 , see also Weierstrass's theo- 
rem 

Infinite products: 22, 10, 129, 57, 
194, exs. 2 and 3 ; uniform conver- 
gence of, 22, 10 , 129, 37 , see also 
Expansions 

Infinite series : see Series 

Infinity : see Point at infinity 

Inflection, point of : 186, 80 

Integral functions: 21, 8, 127, 57; 
associated: 218, ex. 7; class of: 
132, 58 ; with an infinite number of 
zeros: 127, 57; periodic: 147, 65; 
transcendental : 21, ftn , 00, 42 , 
JJff, 61 , 230, 96 

Integral transcendental functions : 21, 
ftn., ft*, 42 , 136, 61 , 230, 96 

Integrals, Abelian : 193, 82 , #4#, 101 ; 
Abelian, of the first, second, and 
third kind: 244, 101; Abel's theo- 
rem: 244, 102; Cauchy's: 76, 33, 
change of variables in: 62, 26; along 
a closed curve: ##, 28; definite: 



see Definite integrals; diffeientia- 
tion of. 77, 33, 227, 95; double: 
see Double integrals; elliptic: 120, 
56 , 174, 78 , 250, 103 ; of elliptic 
functions: 168, 75; fundamental 
formula of the integral calculus: 
63, 26 , 72, 31 ; Hermite's formula : 
215, 90; Hei mite's formula foi el- 
liptic: 165,1%, 168,75, 195, sx 9, 
hyperelhptic : 116, 55, 247, 103; 
law of the mean (Weierstrass, Dar- 
boux): 64,27, line: &Z, 25, 62, 26, 
74, 32, 224, 93; of rational func- 
tions: 33, 15, -WS, 53; of senes: 
86, 39; uniform convergence of: 
229, 96 ; see also Cauchy's theorems 

Invariants (integrals) . 57, ex 15 ; of 
elliptic functions : 158, 70 , 172, 77 , 
188, 79 

Inverse functions: see Functions, in- 
verse, implicit 

Inversion : 45, 19 , 57, exs 13 and 14 

Irrational functions: 13, 6, see also 
Functions 

Irreducible polynomial : 240, 100 "* 

Isolated singular points: 89, 40 , 132, 
59 ; essentially singular : 91, 42 

Isothermal curves : 54, 24 

Jacdbi : 125, ex 18 , 147, 66 , 154, 69 , 

170, 76 , 180, 78 
Jacobi's theorem : 147, 66 
Jensen : 104, 50 
Jensen's formula : 104, 50 

Kepler: 109, ex , jf#0, ex 26 
Kepler' s equation : 109, ex , 126, ex 27 
JTfem : 0, ex 23 

Lagrange : ./0, 51 , 1##, ex. 26 , 251, 

104 
Lagrange's formula: 106, 51, .?<?, 

ex. 26 ; extension of : 250, 104 
Laplace: 10, 3, 54, 24, 55, Note, 

106, 51 , j?5, ex 19 , 251, 104 
Laplace's equation : 10, 3 , 54, 24 , 55, 

Note 
Laurent: 75, 33, AT, 37, 91, 42, 04, 

43, JW, ex.23, 146,6$ 



INDEX 



257 



Laurent's series: 75,33, SI, 37, 146, 

65 

Law of the mean for integrals . 64, 27 
Legendre : 106, ex , 125, ex 18 , 180, 

78 
Legendre's polynomials: 108, ex ; 

Jacobi's f orm: 125, ex 18; Laplace's 

form : 125, ex 19 
Limit point : 90, 41 
Line integrals: 61, 25, 62, 26, 74, 

32 , 84, 93 
Linear tiansformation : 59, ex 23; 

general : 44, ex 2 

Lines, singular: see Natural bound- 
aries, and Cuts 
Lwumlle: <W, 36, 150,51 
Liouville's theorem : 81, 36 , 150, 67 
Logarithmic critical points : 32, 14 , 

118, 53 
Logarithms : 28, 13 , H5, 53 ; natural 

or Napieuan: 28, 13; series for 

Log (1 + ) : 38, 17 
Loops: 118, 53, 115, 54, #44, 101 
Loxodromic curves : 53, ex 1 

Maclaunn: 83, ex 

Maps, conf oimal : 42, 19 , 45, 20 , 48, 

20 , 52, 23 ; geographic: 5#, 23 , see 

also Projection 
Meray : 81, ftn. , 200, f tn 
Mercator's projection . 52, ex 1 
Meromorphic functions : 90, f tn. 
Mtitag-Leffler: 127, 57 and ftn , J734, 

61 , 1S9, 63 
Mittag-Leffler's theorem : 127, 57 , 

1S4, 61 , 155, 63 ; Cauchy's method : 

139, 63 

Monodromic functions : 17, ftn. 
Monogemc functions : 9, ftn. 
Mor&ra : 7S, 34 
Mor era's theorem : 78, 34 
Multiform functions: 17, ftn 
Multiple-valued functions : 17,1 

Napier i 28, 13 

Napierian logarithms : 28, 13 

Natural boundary : 201, 84 , 208, 87 , 

211, 88 
Natural logarithms : 28, 13 



Neighborhood: &, 40; of the point 
at infinity: 109, 52 

Odd functions : 154, 68 

Order, of elliptic functions : 150, 68 ; 

of poles: 89, 40; of zeros: 88, 40 
Ordinary point : 88, 40 

P function, p(u): 154, 68, 182, 79; 
defined by invariants: 182, 79; le- 
lation between p (u) and p' (u) : 15#, 
70 

Pamleve*: 85, 38 

Parabolic transformation : 57, ex. 15 
Parallelogram of periods : 150, 67 
Parametnc representation: see Curves 
Periodic functions . 14 5, 65 ; doubly : 
145, 65, 149, 67, see also Elliptic 
functions 

Penodic integral functions : 147, 65 
Periods : of ctnx : 144, Note 3; cyclic : 
244, 101 ; of definite integrals: 112, 
53 , 1 14, Note ; of elliptic functions : 
158, 68 , 172, 77, 184, 79; of elliptic 
integrals: 180, 56; of functions: 
145, 65 ; of hyperelhptic integrals : 
116, 55; independent: 147, 66; 
parallelogram of: 150, 67; polar: 
118, 53 , 119, 55 , #44, 101 ; primi- 
tive pair of: 149, ftn ; i elation be- 
tween periods and invariants: 172, 
11 ; of sin x : 143, Note 1 
Ptcardi 21, ftn., 93, 42 , 127, ftn. 
Poincarfi 208, ftn , , ftn., #&, 

ftn 

Point, critical or branch : see Critical 
points; double: 184, 80, 191, 82; 
at infinity: 109, 52; of inflection: 
186, 80; limit: 90, 41; ordinary: 
88, 40; symmetnc: 5, ex 17, see 
also Neighborhood, Singular points, 
and Zeros 

Polar periods : see Periods, polar 
Poles: 88, 40, 90, 41, 133, 59; of 
elliptic functions: 150, 68, 154, 
68; infinite number of: 135, 61; 
137, 62 ; at infinity : 110, 52 ; order 
of: 0,40 
Polynomials, irreducible : 240, 100 



258 



INDEX 



Power series : 18, 8 , 196, 83 ; con- 
tinuity of : 7, 2 , 56, ex 7 ; deriva- 
tive of : 19, 8 ; dominating C?l, 9 ; 
lepresentmg an analytic function: 
20, 8 , see also Analytic extension, 
Circle of convergence, and Senes 
Pi imary functions, Weieisti ass's : 127, 

57 

Primitive functions : S3, 15 
Primitive pan of penods : 149, ftn 
Pimcipal part: 89, 40, 91, 42, 110, 

52 , 133, 59 , 135, 61 
Principal value, of arc sm z: SI, 

ftn 

Products, infinite : see Infinite products 
Projection, Mercator's: 52, ex 1; 

stereographic : 53, ex 2 
Puiseux : 240, ftn 

Quantity, imaginary or complex : 3, 1 
Quartics: 187, 81; bicncular: 193, ex 

Rational fraction : 1S3, 59 

Rational functions: 12, 5; integrals 
of : 33, 15 ; of sin z and cos z : 35, 16 

Region, connected : 11, 4 

Regular functions: see Functions, 
regular 

Representation, conformal: see Con- 
formal repiesentation; parametric: 
see Curves 

Residues: 75, 33, 94, 43, 101, 48, 
110, 62, 112, 53; ot elliptic func- 
tions: 151, 68; sum of: 111, 52; 
total : 111, 52 

Rhumb lines : 53, ex. 1 

Riemann : 10, ftn , 50, 22 ; 74, 32 , 
#44, 101 

Riemann surfaces : 244, 101 

Riemann's theorem : 50, 22 

Roots of equations: see Equations, 
D'Alembert's theorem, and Zeros 

Sauvage : 2S1, 97 

Schroder : #14, 89 

Senes, of analytic functions : 86, 39 ; 
AppelPs: 84, 38; Burman's: l#tf, 
ex 26; the Caucby-Laurent : 81, 
35; the Cauchy-Taylor : 79, 35 ; for 



ctn x: 143, 64; differentiation of: 
88, 39; dominant: SI, 9, 157, 69; 
double : see Double senes , integia- 
tion of : 86, 39 ; Laurent's : 75, 33 , 
81, 37, 146, 65; for Log (1 + z) : 
38, 17 ; of polynomials (Pamleve*) : 
86, 38; for tan z, etc : 154, ex 4; 
Taylor's: SO, 8, 75, 33, 78, 35, 
#0#, ftn , 826, 94; umfoimly con- 
vergent : 7, 2 , 86, 39 , 55, 39 , see 
also Lagrange's formula, Mittag- 
Leffler's theorem, and Power series 

Seveial variables, functions of. 218, 
91 , see also Analytic functions of 
several variables 

Sigma function, <r(u) : 162, 72 

Single-valued analytic functions : 127, 
57; with an infinite number of 
singular points, Mittag-Leffler's the- 
orem: 1^4,60; (Cauchy's method) : 
139, 63; with an infinite number 
of zeros, Weierstrass's theoiem: 
128, 57; primary functions: 107, 
57 

Single-valued functions: 17, 7, 127, 
57 

Singular lines: see Cuts and Natuial 
boundaries 

Singular points: 13, 5, 75, 33, 88, 
40, 204, 85, 232, 97; algebraic: 
241, 100 ; on circle of convergence : 
202, 84 and ftn; essentially: 91, 
42; essentially, at infinity: 110, 52; 
infinite number of: 134, 60, 139, 
63; isolated: 89, 40, 132, 59; log- 
arithmic: #44, 101; order of: 89, 
40 ; transcendental : 248, Note , see 
also Critical points, Mittag-Leffler's 
theorem, and Poles 

Singularities of analytical expressions: 
213, 89 , see also Cuts 

Stereographic projection : 53, ex. 2 

StteUjes: 10$ ex. 

Symmetric points : 58, ex. 17 

Systems^ conjugate isothermal : 54, 24 

Tannery : 214, 89 

Taykr: 20, 8, 75, 33, 78, 35, 206, 
ftn., 226, 94 



INDEX 



259 



Taylor's formula, series : 20, 8 , 75, 
SB , 78, 35 , 806, ftn ; for double 
series : 226, 94 

Theta function, 9 (u) : 170, 76 

Total lesidue: 111, 52 

Tianscendental functions: see Func- 
tions 

Transcendental integral functions see 
Integral tianscendental functions 

Transformations, bnational: 192, 82, 
252, ex 1; circular: 45, 19, 57, ex 
13, conformal. 42, 19, 45, 20, 48, 
20, 52, 23; elliptic: 57, ex. 15; 
Fuchsian: 57, ex 15; general hn- 
eai : 44, ex 2 ; hypeibolic : 57, ex 
15 ; mveision : 45, 19 , 57, exs 13 and 
14 ; linear : 59, ex 23 ; parabolic . 57, 
ex 15 , see also Projection 

Tiigonometnc functions: 26, 12; in- 
verse : SO, 14 ; inverse sine . 11 4, 
54, penod of ctn x: 144, Note 3; 
peiiod of sin x : 148, Note 1 ; prin- 
cipal value of: 31, ftn ; lational 
functions of sin z and cos z : 35, 16 , 
see also Expansion 

Umcuisal cuives: 191, 82 



Uniform convergence: see Conver- 
gence, uniform 

Uniform functions : 17, ftn 

Unifoimly conveigent senes and prod- 
ucts : see Conveigence, uniform 

Variables, complex: 6, 2; infinite 
values of* 109, 52; several: see 
Analytic functions of several vari- 
ables 

Wezerstrass*. 64, 27, 88, ftn., 92, 42, 
121, 56 , 187, 57 and ftn , 139, 63 , 
149, 67 , 154, 69 , 156, 69 , 800, ftn , 
212, 88 , 283, 98 , 237, ftn 

Weierstrass's formula: 64, 27, 131, 
56; primary functions: 7^7, 57; 
theorem : 92, 42 , 127, 57 ; J?#, 62 , 
139, 63 , 233, 98 

Zeros, of analytic functions : 88, 40 , 
884, 98 , #4.T, 100 ; of elliptic func- 
tions : 152, 68 , 154, 68 ; infinite 
number of : 26, 11 , 93, 42 , 188, 
57; order of: , 40, see also 
D'Alembert's theorem 

Zeta function, f (u) : 159, 71 



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